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Residential parking in vibrant city districts
Inga Molenda n, Gernot SiegWWU Münster, Institut für Verkehrswissenschaft, Am Stadtgraben 9, 48143 Münster, Germany
a r t i c l e i n f o
Article history:Received 14 January 2013Received in revised form6 February 2014Accepted 17 February 2014
JEL classification:R41R48D61D72
Keywords:Residential parkingUrban vitalityLove of varietyLocal decision-making
a b s t r a c t
Living downtown can be advantageous because it enables convenient access to a variety of shopping andleisure activities, but a major disadvantage is the difficulty of finding a parking spot. We formally modelthe trade-off between privileged parking for residents and economic vitality in terms of the productvariety available in a vibrant city district and identify situations in which assigning on-street parkingspaces to residential parking constitutes an optimal policy, both from a welfare and a residentperspective. However, we demonstrate that privileged parking for residents is unlikely to result in afirst-best allocation of on-street parking spaces, if an efficient level of economic vitality is to be ensuredat the same time.
& 2014 Elsevier Ltd. All rights reserved.
1. Introduction
Downtown areas and other vibrant districts of European citiesare often both commercial and residential areas. Residents, avariety of retail stores and restaurants, and visitors from outsidethe district all add to the vibrancy. For car drivers, the downside ofliving in, working in, or visiting such a district is the struggle tofind an individually suitable parking spot. Because many down-town areas and their surrounding districts of European cities weredeveloped when car ownership was not nearly as common as it isnowadays, residents often lack sufficient private parking capacity,so that they are dependent on public parking spaces. Residentsusually experience a high disutility from searching for a parkingspot in “their” neighborhood and from possibly not being able topark close to their homes, so they often favor parking regulationsthat privilege them. However, residents of vibrant city districtsnormally also enjoy the variety of stores and restaurants in theirneighborhood and they know that non-resident customers arerelevant to the variety offered and that parking policies such as theestablishment of resident parking areas can influence their visitsnegatively.
Still and Simmonds (2000) report results from both attitudinalstudies and land-use/transport models supporting the argumentthat the economic vitality of urban centers is sensitive to theprovision of parking. They emphasize the concerns local autho-rities often have when deciding on parking policies: retailing isimportant to local residents, and maintaining the economicvitality of urban centers also depends on shoppers from outsideas well.
A non-resident visits a vibrant city district to shop, if theassociated private benefit exceeds the associated private cost. Ina setting where more shoppers add to variety that also benefitsother people, but also induce a parking cost increase, due to morecompetition for a suitable parking spot, either too many or too fewnon-residents might visit the district at an unregulated (free on-street parking) equilibrium, depending on the magnitude of theoverall external effect.
Assuming the absence of further market distortions, the avail-able parking capacity is allocated in such a way that the aggregateparking costs of those who park in the district are minimal and thenumber of non-resident shoppers is optimal in the first bestsolution. We will show that parking fees/subsidies and self-selection may lead to the first best solution in a model with twotypes of consumers (residents and non-residents) and two parkingfacilities (on-street and a parking lot). For several reasons, how-ever, the first best solution might not be feasible. Parking subsidiesmay provoke undesired behavior, such as visits by non-residents
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journal homepage: www.elsevier.com/locate/ecotra
Economics of Transportation
http://dx.doi.org/10.1016/j.ecotra.2014.02.0022212-0122 & 2014 Elsevier Ltd. All rights reserved.
n Corresponding author. Tel.: þ49 251 83 22997.E-mail addresses: [email protected] (I. Molenda),
[email protected] (G. Sieg).
Economics of Transportation 2 (2013) 131–139
wishing to earn the subsidy, but with no intention to shop.Furthermore, self-selection through second degree price differen-tiation may be impossible if the residents and/or the non-residentsdiffer in their preference for product variety and their cost of usingthe available parking facilities. Besides, if it is optimal thatresidents pay for parking in theory, it is unlikely that they do soin reality, either because of their lobbying efforts or because ofurban development plans.
If residents have privileges on public parking capacities, muni-cipalities apply some kind of a residential parking policy. InGermany, for example, there are basically two different residentialparking policies: the residents are either exempted from payingthe usually charged parking fees or they are exclusively entitled touse a specific share of on-street parking spaces, such as all parkingspaces on one side of the road. In both cases, the residents need aresidential parking permit that is issued by the municipal roadtraffic departments for an administrative fee of about €30 perannum.1 Important to note is that such a permit gives a specialparking right to the holder, but does not guarantee that there willbe a parking space available.
In a partial equilibrium analysis, we discuss a residentialparking policy according to which a certain share of on-streetparking spaces is reserved for residents as an alternative to thefirst best policy that might prove elusive. We reveal under whichcircumstances assigning on-street spaces to residential parkingwould be rational in principle. Furthermore, taking into accountthat there is a trade-off between parking privileges for residentsand economic vitality in terms of the product variety offered andvalued by residents and visitors, we determine the optimal shareof residential parking spaces. Additionally, because parking poli-cies are decided on a local level and local voters are residents(Arnott, 2011), we further analyze the optimal share of on-streetspaces allocated to residential parking from the resident perspec-tive only and we find that it exceeds what is optimal from thewelfare perspective. With regard to meeting the two objectives ofminimizing the aggregate parking costs of those who park in thedistrict and ensuring an optimal number of non-resident shoppers,such a residential parking policy is certainly inferior to the firstbest policy.
Since transport economists recognized that parking is a crucialelement of urban transportation, parking has received increasingattention in the economic literature. Willson (1995) and Shoup(1999, 2005) discuss planning standards such as minimum parkingrequirements with regard to urban sprawl, automobile use, andthe accompanying social costs. Furthermore, many publicationsaddress cruising for on-street parking in downtown areas, both inisolation and in the context of general traffic congestion as well asboth in the absence and in the presence of an private off-streetmarket (e.g., Glazer and Niskanen, 1992; Arnott and Rowse, 1999,2009; Anderson and de Palma, 2004; Shoup, 2005; Calthrop andProost, 2006; Arnott and Inci, 2006). These studies recommendparking fees that reflect the social cost of parking as an efficientsolution, at least if there is no off-street market. In the presence ofan off-street market, adjusting the on-street parking fee to the off-street price is found to be beneficial in case of an inelastic parkingdemand, since it reduces cruising for parking. This positive effect isempirically observed by van Ommeren et al. (2012) for theNetherlands, where parking fees on- and off-street are quitesimilar. To overcome the opposition of different parties that ariseswhen the introduction of, or an increase in, on-street parking feesis discussed, Shoup (2005, p. 398) proposes the implementation of
benefit districts where the parking revenue “is spent to clean thestreets, plant street trees […] and ensure public safety.” In thiscontext, he also addresses residential parking by contrasting theestablishment of pure residential parking districts with a parkingpolicy that “taxes foreigners living abroad”, while residents parkfor free. van Ommeren et al. (2011) emphasize the inefficienciesthat can result from such a policy. For the residents of Amsterdam,they estimate a marginal willingness to pay of about €10 per dayfor an on-street parking permit and find that it exceeds the actualtariff for a permit considerably, but that it is lower than theparking fee that non-residents pay, which implies an inefficientuse of parking spaces. In a recent contribution, van Ommeren et al.(2014) estimate that each underpriced residential parking permitin fact induces an annual welfare loss of about €275 in Dutchdowntown shopping districts. They state that the welfare lossmostly results from a loss in non-residents' consumer surplusbecause non-residents with a higher willingness to pay for on-street parking than residents are crowded out to more expensivegarage parking spaces. To the best of our knowledge, however,residential parking has not yet been analyzed in the context of thetrade-off between the minimization of aggregate parking costs ofthose who park in the district and the value attached to productvariety offered in a city district that accommodates both residentsand different types of businesses.
2. Model
We consider a city's vibrant residential and commercial districtlocated at 0 on a [0,1] interval. The residents are homogeneous andtheir number is fixed and normalized to 1. The number of storeslocated in the district is denoted by s. Outside the district, anumber of non-residents equal to the number of residents livesuniformly distributed on a [0,1] interval (Fig. 1).
Stores and product variety: Each store offers one variant of adifferentiated product. Although products are heterogeneous, weassume that each store sells a unit of its product at the exogen-ously given price τ. We further assume that the stores have ahomogeneous cost structure. Marginal cost is zero, but each storeincurs an entry cost ϵ¼ EðsÞ, which rises the more stores enter thedistrict (ϵ0 ¼ dE=ds40), at either a constant or an increasing rate(ϵ″Z0). Such an assumption can be justified by the district'slimited spatial capacity and the ensuing difficulties in finding anadequate location the more stores enter.
Both residents and non-residents value product varietyand each resident and each non-resident who visits the districtbuys one unit of each product offered. The number of visitingnon-residents is denoted by v, so that the profit function of eachstore i is
Π ¼Π iðv; EðsÞÞ ¼ τ � ½1þv��EðsÞ: ð1ÞA store i operates in the district if ΠZ0. For the marginal entrant,Π ¼ 0 holds, so that there is no more entry as soon as τ �½1þv� ¼ EðsÞ ¼ ϵ applies. By means of the inverse function ofϵ¼ EðsÞ, s¼ E�1ðϵÞ, we find the zero-profit number of stores tobe a function of the number of visitors
sðvÞ ¼ E�1ðϵÞ ¼ E�1ðτ � ½1þv�Þ ð2Þwith ds=dv¼ τ � ½E�1�0 ¼ τ=E040 and d2s=dv2 ¼ �τ � E″=½E0�3r0.
0 1
residentsstores
non - residents
Fig. 1. Spatial model structure.
1 Applicants for residential parking permits have to meet a number ofrequirements. For example, they normally have to be the owner of the car forwhich the permit is valid, they can apply for one permit only or they must not havea private parking space.
I. Molenda, G. Sieg / Economics of Transportation 2 (2013) 131–139132
The net utility that a resident or a visiting non-resident gainfrom shopping or, more generally, from the variety offered in thedistrict is ~ujðsÞ with j¼ r;nr, where the subscript r refers to aresident and the subscript nr refers to a non-resident.2 We assumethat ~ujð0Þ ¼ 0 and that the marginal utility gained from the varietyprovided by an additional store is constant or decreasing positive;thus it is d ~uj=dS40 and d2 ~uj=dS
2r0. Furthermore, as the numberof stores, and with that the product variety, increases when anadditional non-resident decides to visit the district, his or herdecision involves a positive external effect. Thus, formally, it is~ujðsðvÞÞ with d ~uj=dv¼ d ~uj=ds � ds=dv40.
Travel cost: When traveling to the district, non-residents bearper-unit car travel cost of t. We assume that non-residents have noalternative to traveling by car and that they do not carpool, buttravel alone.
Parking capacity and costs: We assume that each resident ownsa car, but does not have a private parking space. Thus, bothresidents and visiting non-residents have two possible ways topark their cars in the district: either on-street in direct proximityto the residential houses and stores or in a public parking lot,which is assumed to be slightly further away from the residentialhouses and stores.3 Because of that, residents and visitors do notconsider the parking options as perfect substitutes. In fact, weassume that a driver j¼ r;nr does not face costs when parkingon-street, but that a resident bears a cost of cr and a non-residentwho visits the district a cost of cnr when parking in the lot. Thesecosts cr and cnr are usage costs and comprise the driving cost to theparking lot and the walking cost to the residential houses andstores and back again (Fig. 2).
We assume that the provision of the parking capacities doesnot entail costs. The number of on-street parking spaces equals thenumber of residents, and hence the number of cars owned byresidents, thus xo¼1. For the parking lot's capacity xl, we assumexlZ1. Both capacities combined, there is sufficient parking spacefor all drivers with a demand for parking. We do not account for atemporal variation in demand, but assume that residents andvisitors arrive at the district at the same time. In the absence ofany parking regulations, a driver j¼ r;nr finds an on-street spaceby a random-rationing rule, as in Calthrop and Proost (2006). Theprobability of finding a vacant on-street space is
ρ¼ xo1þv
¼ 11þv
: ð3Þ
We assume that the municipality has implemented an efficientparking guidance system and that there is sufficient road capacity.Thus, cruising for parking (e.g., Glazer and Niskanen, 1992; Arnottand Rowse, 1999, 2009; Anderson and de Palma, 2004; Shoup,2005; Arnott and Inci, 2006) or the accompanying social costs isnot considered in our model.
As there are just as many on-street parking spaces as residents,a resident has a ρ¼ 1 probability of finding an on-street spot in thecase of v¼0. However, rivalry in consumption emerges if v40.In this case, each driver j¼ r;nr in the district faces a 1�ρ40probability of having to park in the lot of and expects a parkingcost of
pcjðvÞ ¼ cj � ½1�ρ� ¼ cj �v
1þv
� �with
dpcjdv
¼ cj½1þv�240: ð4Þ
Thus, the visit of an additional non-resident not only involves apositive, but also a negative external effect, since it leads to anincrease in both the parking cost that a resident and a visitorexpects.
In what follows, we discuss two parking regimes, namely theregimes pf (for “parking fees”) and rp (for “residential parking”)with regard to their ability to ensure both a first-best allocation ofon-street parking spaces and an efficient number of visitors vn. Theregime pf constitutes the benchmark, but our focus in this paper lieson regime rp. For reasons of clarity and comprehensibility, theexplicit design of each parking regime k¼ pf ; rp is directlyexplained before its analysis. Note at this point only that, dependenton the applied parking regime k, each resident incurs a full parkingcost of pcr;k and each visitor of pcnr;k.
Overall and aggregate utilities: In summary, each residentreceives an overall utility from living in the vibrant district ofur;k ¼ ~urðsðvÞÞ�pcr;k and a non-resident who lives at wA ½0;1�receives an overall utility from visiting the vibrant city district ofuw;k ¼ ~unrðsðvÞÞ�t �w�pcnr;k. We assume that non-residents gainan alternative utility of zero when they do not visit the district.Thus, non-resident w visits the district if uw;kZ0.
For the analysis of the different parking regimes k, we need theaggregate utilities. The residents receive an aggregate utility of
Ur;k ¼ r � ur;k ¼ 1 � ½ ~urðsðvÞÞ�pcr;k� ð5Þand the visitors, and simultaneously all non-residents, of
Unr;k ¼Z v
0uw;k dwþ
Z 1�v
v0¼ ~unrðsðvÞÞ � v�
t2� v2�pcnr;k � v: ð6Þ
Adding up Ur;k and Unr;k gives welfare
Wk ¼ ~urðsðvÞÞ�pcr;kþ ~unrðsðvÞÞ � v�t2� v2�pcnr;k � v: ð7Þ
3. Parking fees
3.1. Optimal fees/subsidies
Optimal parking fees should both allocate on-street parkingspaces to the group with the higher cost of using the parking lotand subsidize or tax the visitors to remedy the possible marketfailure of excess or insufficient entry. Thus, the parking feesensure that the explained externalities of an additional visitorare taken into account. Suppose that under parking regime k¼pf,the municipality charges the lump sum fees fo for on-streetparking and fl for parking in the lot, which can be interpreted assubsidies if they are negative. With regard to the cost of using theparking lot, we first consider case (a) that a resident bears a highercost than a non-resident and proceed then with the reversecase (b).
In case (a), it is cr4cnr and optimal when residents park on-street and visitors use the parking lot. Thus, the fees foa and fl
a haveto ensure that (i) visitors prefer to park in the lot, (ii) residentsprefer to park on-street and (iii) the pricing system leads to anumber of visitors vn that maximizes welfare. Self-selection (i) isensured if pcanr;pf ¼ cnrþ f al o f ao and self-selection (ii) if pcr;pf ¼f aorcrþ f al , which implies that both (i) and (ii) are ensured if
residents
stores
on-street parking
parking lot
Fig. 2. Structure of city district located at 0.
2 We neglect τ as a function argument, since τ is exogenous.3 Employees working at the districts' different stores are assumed to demand
neither on-street nor off-street parking.
I. Molenda, G. Sieg / Economics of Transportation 2 (2013) 131–139 133
f ao ¼ crþ f al .4 Hence, charging the fee fo
a prevents the negativeexternality in terms of increased expected parking costs that anadditional visitor would cause in the absence of any regulations, asit is clear who parks where from the beginning. The fee (subsidy)fla that the visitors pay (receive) for parking in the lot should beused to ensure the optimal number of visitors vn. The parking feesare expenditures for those who park in the district, but revenuesfor the municipality (or the converse) and therefore do not changewelfare. In this case, welfare is
Wa ¼ ~urðsðvÞÞþ ~unrðsðvÞÞ � v� t2� v2�cnr � v: ð8Þ
The optimal number of visitors vn is obtained when dWa=dv¼ 0,or when
d ~ur
dsþv � d ~unr
ds
� �dsdv
þ ~unrðsðvÞÞ ¼ t � vþcnr : ð9Þ
However, the number of visitors cannot be determined directly bya social planner, but results from the individual decisions of non-residents. Recall that non-residents receive a zero alternativeutility if they do not visit the district, thus non-resident wA ½0;1�visits the district in case (a) if
uaw ¼ ~unrðsðvÞÞ�t �w�pcanr ¼ ~unrðsðvÞÞ�t �w�cnr� f al Z0: ð10Þ
For the indifferent non-resident w¼v, who at the same timedefines the total number of visitors, condition (10) is binding, sothat
f al ¼ ~unrðsðvÞÞ�t � v�cnr ð11Þmust hold. Inserting (9) (into 11) yields
f al ¼ � d ~urðsðvnÞÞds
þvn � d ~unrðsðvnÞÞds
� �dsðvnÞdv
; ð12Þ
which describes the negative parking fee or the parking subsidythat each non-resident receives when visiting the district andwhich is equivalent to the positive externality of the visitor whomaximizes welfare. Thus, by subsidizing each visitor with fl
a and bycharging each resident the fee f ao ¼ crþ f al , the first best solutioncan be achieved.
If case (b) applies, thus crocnr holds as suggested by vanOmmeren et al. (2011), it is optimal if visitors park on-street andall on-street parking spaces are utilized. Therefore, the fees (orsubsidies) fob and fl
b must induce (i) visitors to park on-street and(ii) residents to be indifferent between the two options. Self-selection is ensured if pcbnr ¼ f bo ¼ crþ f bl ¼ pcbr , which looks similarto the self-selection condition in case (a), but here it implies thatvisitors park on-street instead of in the lot. The fee fo
b should beused to ensure the optimal number of visitors vn. Welfare in thiscase is
Wb ¼ ~urðsðvÞÞþ ~unrðsðvÞÞ � v�t2� v2�cr � v: ð13Þ
The welfare functions Wa and Wb, given by Eqs. (8) and (13), aredistinguished by the last term: in the case of cr4cnr , an additionalvisitor parks in the lot and therefore reduces welfare by cnr whilehe or she parks on-street in the case of crocnr and thereby crowdsout a resident who then uses the parking lot at the cost of cr.
The optimal number of visitors vn is obtained when dWb=dv¼ 0,or when
dWb
dv¼ d ~ur
dsþv � d ~unr
ds
� �dsdv
þ ~unrðsðvÞÞ�cr ¼ t � v: ð14Þ
Again, whether or not a non-resident visits the district is his or herindividual decision. In case (b), non-resident wA ½0;1� gains an
overall utility of
ubw ¼ ~unrðsðvÞÞ�t �w�pcbnr ¼ ~unrðsðvÞÞ�t �w� f bo ð15Þ
that is equal to zero for the indifferent visitor w¼v. Thus,
f bo ¼ ~unrðsðvÞÞ�t � v ð16Þ
must hold and inserting (14) (into 16) yields
f bo ¼ � d ~urðsðvnÞÞds
þvn � d ~unrðsðvnÞÞds
� �dsðvnÞdv
þcr ; ð17Þ
which is the optimal on-street parking fee (or subsidy). In this case,the fee that ensures vn not only comprises the positive, but also thenegative externality of the visitor who maximizes welfare. Those ofthe residents who park in the lot still experience the usage cost of crbut receive f bl ¼ f bo�cr , thus an amount of the positive externalitythe welfare-optimizing visitor causes, by which their full cost ofparking is reduced.
3.2. Discussion
In our model, we assume that there are two homogeneoustypes of car driver (residents and visitors) and two parkingpossibilities (on-street and a parking lot) in the district. Therefore,second degree price discrimination is possible. In reality, however,such a parking regime may be difficult to implement. First of all, itis more likely that the visitors are heterogeneous with regard toboth their preference for product variety and the cost of using theparking lot. Hence, it is almost impossible to efficiently tax/subsidize their visits by means of on- or off-street parking fees,especially when these are also used as an instrument for allocatingthe on-street parking spaces to drivers who incur the highest costof using the parking lot. This objective becomes even moredifficult to achieve if the residents are also heterogeneous regard-ing the cost of using the lot. Secondly, there could be more thanonly two parking possibilities in the district and its surroundings,all of which have different usage costs, so that minimizing themand ensuring an efficient number of customers from the outside isdifficult or even impossible.
Moreover, as some of the parking fees are or may constituteparking subsidies, parking costs may become parking revenuesfor a driver and this might have undesired effects which requirefurther regulation. For example, people from outside may visitthe district without any intention to shop, but to earn thesubsidy.5 Furthermore, if there were residents who had pre-viously used their private parking space, they could now havean incentive to use public parking instead. More generally, carownership could become more attractive for residents and there-fore increase.
If the optimal on-street parking fees are positive, it is verylikely that residents of downtown areas or other vibrant citydistricts would oppose such a policy in “their” neighborhood.Basically, of course, residents do not automatically acquire parkingprivileges in case the on-street parking capacity is public. Nor-mally, however, the municipalities pursue parking policeswhereby residents do not have to pay for parking, whether dueto the residents' lobbying efforts or due to urban developmentplans. In Germany, such plans aim at making city center districtswith inherently scarce parking capacity attractive places of
4 To keep the notation as simple as possible, we drop the index pf in thefollowing analysis of optimal parking fees for all relevant welfare measures.
5 To the best of our knowledge, parking subsidies as discussed above are notobservable in reality and their theoretical efficiency depends substantially on ourassumptions (no cruising for parking, provision of parking capacities does notentail any (opportunity) costs). However, what we do observe, at least in Germany,is that some businesses refund all or part of their customers' parking expenses ifthey present their parking ticket on checking out.
I. Molenda, G. Sieg / Economics of Transportation 2 (2013) 131–139134
residence, especially for families of a certain income group thatotherwise would settle in suburbs.
As soon as municipalities confer special parking rights onresidents, they pursue a residential parking policy. A policy thatexempts residents from paying the parking fees, but that fulfillsthe two objectives – minimization of parking costs and ensuringan efficient number of visitors – is difficult to achieve. In case (a),self-selection (ii) may not work. Dependent on the magnitude ofthe parking subsidy that visitors receive for parking in the lot, itcould also pay off for residents to park there. In case (b), self-selection (ii) only works if residents receive a subsidy equal to thecost of using the parking lot. Unless visitors can effectively bedistinguished from residents, this subsidy has to be consideredwhen determining the on-street parking fee that visitors face andthat ensures self-selection (i). But then, this on-street fee mayprevent the optimal number of visitors in the district.
4. Residential parking
4.1. Parking regime
Because of the difficulties associated with a residential parkingpolicy according to which the municipality exempts residents frompaying the parking fees, but tries to meet the two above-mentioned objectives, we discuss a policy according to whichthe municipality reserves a certain share of on-street parkingspaces exclusively for residents, but does not charge parking feesat all. In principle, the reservation of on-street parking spaces forresidents then minimizes the aggregate parking costs of those whopark in the district only if residents bear a higher cost of using theparking lot than non-residents, that is if cr4cnr holds, and if allon-street parking spaces are reserved for residents. This, however,implies the subsidization of non-resident visits, and subsidies canprovoke undesired behavior. To encourage non-residents to visitthe district without paying them a parking subsidy, it might beoptimal to reserve some, but not all, on-street spaces for residents.However, in this case neither are the aggregate parking costsminimal nor is the optimal number of visitors ensured, so thatsuch a residential parking policy is inevitably inferior to one thatincludes parking fees/subsidies for all those who park in thedistrict.
Recall that in the case of unregulated parking, the probability offinding a vacant on-street spot was ρ¼ 1=ð1þvÞ. Rivalry in con-sumption emerged if v40, so each member of both groups faced a1�ρ40 probability of having to park in the parking lot. Underparking regime k¼rp, the municipality assigns the share αA ½0;1�of on-street parking spaces to residential parking only. If α¼ 0,none of the on-street spaces are reserved for residents and if α¼ 1,all of the on-street spaces are reserved for residents. Thus, eachresident has a probability of
ρr ¼1�α1þv
þα¼ 1þα � v1þv
ð18Þ
to find a vacant on-street spot, whereas a non-resident who visitsthe district is able to park on-street with a probability of
ρnr ¼1�α1þv
: ð19Þ
The probabilities of having to park in the lot are the converseprobabilities 1�ρr and 1�ρnr . Hence, under paring regime k¼rpeach resident expects a parking cost of
pcr;rpðv;αÞ ¼ cr � ½1�ρr � ¼ cr � v � ½1�α�1þv
ð20Þ
with
∂pcr;rp∂v
¼ cr � ½1�α�½1þv�2 Z0 and
∂pcr;rp∂α
¼ �cr � v1þv
o0;
whereas each visitor of the district expects a parking cost of
pcnr;rpðv;αÞ ¼ cnr � ½1�ρnr � ¼ cnr � vþα1þv
ð21Þ
with
∂pcnr;rp∂v
¼ cnr � ½1�α�½1þv�2 Z0 and
∂pcnr;rp∂α
¼ cnr1þv
40:
Note that these expected parking costs pcr;rp and pcnr;rp constitutethe full cost of parking a resident or a visiting non-resident bearssince the municipality does not charge parking fees at all.6
4.2. Equilibrium with residential parking
We assume that each resident has a parking permit whichindicates that he or she is entitled to park in the declaredresidential parking spaces. The permit is issued as soon as some-one settles in the district and is free of charge. In reality, the feesfor residential parking permits are very low. In Germany, forexample, the fee for an annual parking permit is about €30,implying a daily cost of about €0:08, which is negligible. For theresidents of Amsterdam, van Ommeren et al. (2011) find amarginal willingness to pay of €10 per day for an on-street parkingpermit. Thus, in theory, each of the homogeneous residents buys aparking permit if the fee does not exceed his or her willingness topay. Possible expenditures for parking permits do not affectwelfare, as they correspond to the revenues earned by themunicipality and are returned in some way. If, however, the priceof a parking permit exceeds each resident's willingness to pay,there is no demand for such permits and an allocation of parkingspaces to residential parking does not occur, as unused allocationsconstitute a waste of space.
To calculate the optimal share of residential parking spaces α,we first determine the equilibrium number of stores se and visitorsve as a function of α.
Under regime rp, a non-resident living at wA ½0;1� gains anoverall utility of uw ¼ ~unrðsÞ�t �w�pcnrðv;αÞ. For the indifferentnon-resident w¼v, uw¼0 holds and defines a function v(s) with
dvds
¼ � ∂uw=∂sjw ¼ v
∂uw=∂vjw ¼ v¼ � d ~unr=ds
�t�cnr=½1þv�240: ð22Þ
Since ~unrð0Þ ¼ 0, a non-resident visits the district only if there is apositive number of stores. Furthermore, it is
d2v
ds2¼ � ∂2 ~unr=∂s2
�t�cnr=½1þv�2r0: ð23Þ
Recall from Section 2 that the number of stores is given by
sðvÞ ¼ E�1ðϵÞ ¼ E�1ðτ � ½1þv�Þwith ds=dv¼ τ � ½E�1�0 ¼ τ=E040 and d2s=dv2 ¼ �τ � E″=½E0�3r0.
Assume that ~unrðE�1ðτÞÞ4α � cnr , ~unrðE�1ð2τÞÞotþcnr � ½1þα�=2and ~u 0
nrð0Þ=½tþcnr=4�oE0ð0Þ=τ . There is then a unique interiorequilibrium 0ose and 0oveo1. The first condition ensures thatthe buying power of the residents leads to such a variety thatvisiting the district pays off for at least some non-residents. Thesecond condition ensures, due to the stores' cost of entry to thedistrict, that the combined buying power of the residents and allnon-residents does not result in a sufficient variety, that all non-residents consider the district worth visiting. The third condition
6 In our further analysis, we drop the index rp for all measures to keep thenotation simple.
I. Molenda, G. Sieg / Economics of Transportation 2 (2013) 131–139 135
ensures that the curve v¼ vðsÞ in Fig. 3, which fulfills the visitors'free entry condition uw¼0, is flatter than the curve s¼ sðvÞ thatfulfills the stores' free entry condition Π ¼ 0.7 The intersection of v(s) and s(v) determines the equilibrium number of visitors ve andof stores se.
At equilibrium, ve and se depend on different parameters likethe parameter that indicates the residents' and the non-residents'valuation of the product variety, the per-unit travel cost, the costthat non-residents incur when they have to use the parking lotand the specific entry cost incurred by each store. Since we focuson a residential parking policy that assigns on-street parkingspaces to residential parking only, we treat all these exogenousparameters as constant, except αA ½0;1�. Thus, we define theequilibrium number of visitors as a function veðαÞ and theequilibrium number of stores as a function seðαÞ.
We use the condition uw ¼ ~unrðsÞ�t �w�pcnrðv;αÞ ¼ 0 for theindifferent non-resident w¼v to calculate
dvdα
¼ �∂uw=∂αjw ¼ v
∂uw=∂vjw ¼ v¼ � �cnr=½1þv�
�t�cnr ½1�α�=½1þv�2o0: ð24Þ
The sign of Eq. (24) entails a downward shift of v(s), as shown inFig. 3. Because ∂Π=∂α¼ 0, s(v) does not shift, thus
dve
dαo0 ð25Þ
holds. In the following analysis, we assume �1odve=dαo0. Sinceds=dv40, it follows that
dse
dαo0: ð26Þ
At the free entry equilibrium, non-resident wA ½0;1�, who visitsthe district, receives an overall utility of
uew ¼ ~ue
nrðseðαÞÞ�t �w�pcenrðveðαÞ;αÞ
¼ ~uenrðseðαÞÞ�t �w�cnr � ½αþveðαÞ�
1þveðαÞ ð27Þ
and each resident receives a utility of
uer ¼ ~ue
r ðseðαÞÞ�pcer ðveðαÞ;αÞ ¼ ~uer ðseðαÞÞ�
cr � ½1�α� � veðαÞ1þveðαÞ : ð28Þ
Lemma 1. If the share of residential parking spaces increases, boththe residents' and the non-residents' utility from product varietydecreases as the number of stores declines; thus it is
d ~uej
dα¼ d ~ue
j
dsedse
dαo0 with j¼ r;nr: ð29Þ
Furthermore, an increase in the share of residential parking spacesresults in an increase in the parking cost a non-resident expects:
dpcenrdα
¼ ∂pcenr∂ve
dve
dαþ∂pcenr
∂α¼ cnr � ½½1�α�ve 0ðαÞþveðαÞþ1�
½1þveðαÞ�2 40; ð30Þ
but to a decrease in the parking cost a resident expects,
dpcerdα
¼ ∂pcer∂ve
dve
dαþ∂pcer
∂α¼ �cr � ½veðαÞþveðαÞ2�½1�α�ve0ðαÞ�
½1þveðαÞ�2 o0:
ð31Þ
4.3. Welfare analysis
In the case of parking regime rp, welfare is
We ¼ ~uer ðseðαÞÞ�pcer ðveðαÞ;αÞþ ~ue
nrðseðαÞÞ � veðαÞ� t2� ½veðαÞ�2�pcenrðveðαÞ;αÞ � veðαÞ: ð32Þ
To find the optimal share of residential parking spaces αn, wederive We with respect to α and obtain, after some rearrange-ments,
dWe
dα¼ d ~ue
r
dsedse
dα�∂pcer
∂vedve
dα�∂pcer
∂α
þve � d ~uenr
dsedse
dα�∂pcenr
∂vedve
dα�∂pcenr;rp
∂α
� �
þdve
dα� ~ue
nr�t � ve�pcenr� �
:
The term ½ ~uenr�t � ve�pcenr � describes the marginal visitor's utility
which equals zero at the visitor's optimum. Thus, we find
dWe
dα¼ d ~ue
r
dsedse
dα�∂pcer
∂vedve
dα�∂pcer
∂α
þve � d ~uenr
dsedse
dα�∂pcenr
∂vedve
dα�∂pcenr
∂α
� �
¼ d ~uer
dα�dpcer
dαþve � d ~ue
nr
dα�dpcenr
dα
� �; ð33Þ
which we set equal to zero.
Proposition 1. The optimal share of residential parking spaces αn isfound when the decrease in both the residents' and visitors' utilityfrom product variety equals the savings in overall expected parkingcost; thus when
d ~uer
dαþve � d ~u
enr
dα¼ dpcer
dαþve � dpc
enr
dα: ð34Þ
To actually realize savings in overall expected parking cost,
dpcerdα
þve � dpcenr
dα¼ ve 0ðαÞ½crþcnr � veðαÞ�½1�α�
½1þveðαÞ�2 þ½cnr�cr� � veðαÞ1þveðαÞ ð35Þ
has to be negative, which is only possible if
crcnr
4 1þ ½1þveðαÞ� � ve 0ðαÞ � ½1�α�veðαÞ½1þveðαÞ��ve 0ðαÞ½1�α�
� �: ð36Þ
Thus, inequality (36) defines a necessary condition for a positiveshare of residential parking spaces to be optimal. Because the rightside of (36) is less than one, visitors may bear a somewhat lowerparking cost in the outside lot than residents. Note, however, that(36) is not sufficient for residential parking to be welfare-enhan-cing, since it yet does not ensure that dWe=dα is positive.
4.4. Local decision-making
In reality, decisions on parking regulations are normally madeat a local level (Arnott, 2011). Business owners may oppose
Fig. 3. Free entry equilibrium.
7 The figure with its three graphs is based on the results of our numericalexample in Section 4.5. To be more specific, we set cnr ¼ 1=5. Furthermore, we setα¼ 1=10 to get v1ðsÞ and α¼ 9=10 to get v2ðsÞ. The behavior of s(v) results directlyfrom our assumptions in the example.
I. Molenda, G. Sieg / Economics of Transportation 2 (2013) 131–139136
regulations that prevent potential customers from visiting thedistrict. In our model, however, we assume that business ownersare non-residents and that businesses are perfectly mobile.We thus omit business lobbying efforts in the analysis. Asresidents are assumed to be homogeneous, individual preferencesequal collective preferences. Thus, we simply derive Ue
r ¼ 1 �½ ~ue
r ðseðαÞÞ�pcer ðveðαÞ;αÞ� with respect to α and set it equal to zeroin order to determine the preferred share of residential parkingspaces. Because
dUer
dα¼ ∂ ~ue
r
∂sedse
dα�∂pcer
∂vedve
dα�∂pcer
∂α¼ d ~ue
r
dα�dpcer
dα¼! 0; ð37Þ
the optimal share of residential parking spaces from the residentperspective αn
r is found when the decrease in the residents' utilityfrom product variety equals the residents' savings in expectedparking cost; thus when
d ~uer
dα¼ dpcer
dα: ð38Þ
Even if there is a ballot and both residents and visitors are entitledto vote, one of the residents is the median voter and, according tothe median voter theorem, αn
r is the adopted policy.
Proposition 2. If local authorities only consider the preferences ofthe district's residents, but ignore the preferences of non-residents aspotential visitors when deciding on residential parking, or if there is aballot on this issue, the resulting share of residential parking spacesexceeds the welfare-optimizing share, i.e. αn
r 4αn, whenever thewelfare-optimizing share of residential parking spaces is positive,but less than one.
Proof. See appendix.
Residents prefer a higher share of on-street parking spaces tobe reserved for them than is optimal from the welfare perspective,because they ignore the negative effects of their decision on theoverall visitors' utility in terms of a lower utility from productvariety and a higher expected parking cost.
4.5. A numerical example
The following example illustrates our general results. Weassume that residents and non-residents share the same prefer-ence for product variety. More precisely, ~urðsÞ ¼ ~unrðsÞ ¼ ½μ�τ� � s.Recall that the price each store charges its customers is givenexogenously. We assume that it is τ¼ 1=2. Furthermore, we setμ¼ 1, so that ~urðsÞ ¼ ~unrðsÞ ¼ s=2. The per-unit travel cost a non-resident incurs when visiting the district is t¼1. Altogether, itfollows that
ur ¼ ~urðsÞ�pcrðv;αÞ ¼12� s�cr � ½1�α� � v
1þvð39Þ
and
uw ¼ ~unrðsÞ�t �w�pcnrðv;αÞ ¼12� s�w�cnr � ½vþα�
1þv: ð40Þ
Non-resident wA ½0;1� visits the district if uwZ0. We assume that0ocnro1=4, which implies that even some non-residents visitthe district if all on-street spaces are assigned to residentialparking. For the indifferent non-resident w¼v, uw¼0 holds, sothat the number of visitors as a function of the number of stores is
vðsÞ ¼sþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½s�2�2cnr �2þ8 � ½s�2cnrα�
q�2�2cnr
4: ð41Þ
With regard to the entry cost that each store incurs, we assumeEðsÞ ¼ e � s with e¼1, which defines
sðvÞ ¼ 12 � ½1þv�: ð42Þ
Both v(s) and s(v) are shown in Fig. 3 for cnr ¼ 1=5 and α¼ 1=10, aswell as cnr ¼ 1=5 and α¼ 9=10. Inserting (41) into yields theequilibrium number of stores
se ¼ seðαÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þcnrþc2nr�3cnrα
pþ1�cnr
3ð43Þ
and inserting (43) then into (41) yields the equilibrium number ofvisitors
ve ¼ veðαÞ ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þcnrþc2nr�3cnrα
p�1�2cnr
3; ð44Þ
also shown in Fig. 3.With the explicit results for se and ve, we can determine
the explicit expressions for ~uej ðseÞ and pcej ðve;αÞ with j¼ r;nr, and
thus for dWe=dα and dUer=dα (see Eqs. (33) and (37)). Solving
dWe=dα¼ 0 for α yields
α¼ c2nr ½21þ16cnr�þ3c2r ½7þ16cnr � ½1þcnr ���2crcnr ½29þ8cnr ½1þ4cnr ��16½3cr�2cnrÞ�2cnr
:
ð45ÞThus, the welfare-optimizing share of residential parking spaces is
αnðcr ; cnrÞ ¼0 if crrc
α if cocroc
1 if crZc
8><>: ð46Þ
with
c ¼ cnr ½29þ½20�32cnr �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þcnrþc2nr
pþ8cnr ½1þ4cnr ��
21þ48cnr ½1þcnr �and
c ¼ 16c2nr�7cnr12cnr�3
:
If the decision on the share of residential parking spaces ismade at a local level, we simply have to solve dUe
r=dα¼ 0 for α.We obtain an optimal share of residential parking spaces from theresident perspective of
αn
r ðcr ; cwÞ ¼ð16þ7=cnrþ16cnr�ð6crþcnrÞ=c2r Þ=48 if crr ~c
1 if cr4 ~c:
(
ð47Þwith ~c ¼ cnr=½1�4cnr �.
Fig. 4 depicts our results for αn and αnr when cnr ¼ 1=8. If croc ,
assigning some, but not all on-street parking spaces to residentialparking is only optimal from the resident perspective. If cocro ~c ,however, a social planner also would assign some, but not all ofthe on-street spaces to residential parking, albeit fewer than theresidents would do. From the resident perspective, exclusiveresidential parking in on-street spaces is optimal if ~cocr , whereasa share of residential parking spaces smaller than one is welfare-optimizing as long as croc . Only if crcr would a social planneralso assign all on-street parking spaces to residential parking.
Fig. 4. Optimal shares of residential parking spaces.
I. Molenda, G. Sieg / Economics of Transportation 2 (2013) 131–139 137
In our example, we assumed that residents and non-residentshave homogeneous preferences for product variety for which weused a numerical description. As a result, the optimal shares ofresidential parking spaces seem to be independent of this pre-ference. To show that this is not the case, we allow for the casethat residents and non-residents differ in their preferences forproduct variety. We still assume that ~unr ¼ ½μ�p� � s¼ ½1�1=2� � s,but for the utility a resident derives from product variety, weassume ~ur ¼ ½λ�p� � s¼ ½λ�1=2� � s with λZμ¼ 1. Thus, the utilitya resident receives from product variety is at least as high as thatobtained by a non-resident who visits the district. The calculationsof the equilibrium values are as usual, the only difference beingthat some of them contain the variable λ, as the residents'aggregate overall utility and welfare. Solving dWe=dα¼ 0 for αin this case yields
α¼ ½5�16cnr �c2nr�2cnr ½11þ8cnr ½1þ4cnr ��crþ3½7þ16cnr ½1þcnr ��c2r16cnr ½2cnr�3cr �2
þ½cnr ½7þ8cnr�3λ��9cr�λ4½2cnr�3cr �2
ð48Þ
with
dαdλ
¼ cnr ½7þ8cnr�6λ��9cr4½2cnr�3cr �2
o0; ð49Þ
at least for crZcnr . From the sign of Eq. (49), we can deduce thatthe optimal share of residential parking spaces from the welfareperspective is the lower the more the residents value the varietyoffered in their neighborhood. The same relationship of courseapplies for the optimal share of residential parking spaces from theresident perspective, as the variable λ is part of a resident's utility.We can further infer that there is also a negative relationshipbetween the non-residents' (gross) utility from variety μ and theoptimal share of residential parking spaces from the welfare (butnot the resident) perspective.
5. Conclusion
Many cities provide residential parking permits for residentswho live in downtown areas and in other city districts where on-street parking capacity is scarce. These permits either allowresidents to park for free anywhere in their own neighborhoodwhile non-residents pay for parking, or allow residents to park inon-street spaces that are reserved for their exclusive use. Withinthe context of a formal model of a vibrant city district whoseresidents and visitors appreciate the product variety offered, butmight experience inefficiently high parking costs, we focused onthe latter alternative and analyzed the trade-off between themore-convenient-parking effect on the residents' side, due toresidential parking, and the loss-of-variety effect, due to fewershoppers coming from outside the district. We determined theshare of residential parking spaces that is optimal both from awelfare and from the resident perspective, and found that if thedecision is made locally, more than welfare-optimal parkingspaces are assigned to residential parking. This possible latterresult is problematic insofar as the discussed residential parkingpolicy is already not the first best solution with regard to parkingcost minimization and ensuring an efficient number of non-residentshoppers. A first best solution includes price-discriminated parkingfees (or subsidies), but might be difficult to implement.
Our analysis refers only to one single city district, so that theeffects of the discussed parking policies on the utilities of residentsand other actors in neighboring city districts are not considered.Such an approach is appropriate when decisions on parkingpolicies are made on a district-by-district basis, as may be the
case in large cities that are divided into several administrativedistricts.
In this analysis, we assumed that the provision of on- and off-street parking capacity does not entail any cost and that drivers donot cruise for a parking spot. An important extension of ourresidential parking analysis would relax these assumptions. Withregard to negative cruising externalities, the first best solution canprovide a remedy, whereas a policy of reserving a share of on-street parking spaces for residents is likely to exacerbate theproblem.
A further and ambitious approach for future research wouldentail a public choice analysis, including political lobbying. Weassumed that retail stores, restaurants and other businesses areperfectly mobile and therefore indifferent to parking regulationsthat may have a negative effect on the visits of non-residentcustomers. This assumption rarely holds in reality and therefore,businesses usually oppose parking restraint policies, whichinclude the discussed residential parking policy. Thus, the result-ing parking regime depends on the institutional design that thecity or city district municipality uses to determine its parkingpolicy.
Acknowledgments
We thank Jos van Ommeren, Mogens Fosgerau, two anonymousreferees, Ulrich van Suntum, Aloys Prinz and the participants ofthe 2013 Kuhmo NECTAR Conference for helpful comments andsuggestions.
Appendix A
A.1. Notational glossary
Variables Explanations number of storesv number of visitorsτ price per unit of productϵ entry cost of each store (ϵ¼ EðsÞ)Π profit of each store~ur utility of a resident from product variety~unr utility of a non-resident who visits the district from
product varietyt per unit car travel cost that a non-resident bears
when traveling to the districtxo on-street parking capacityxl capacity of parking lotcr cost a resident bears when parking in the lotcnr cost a visiting non-resident bears when parking in
the lotρ probability of finding an vacant on-street parking
spotpcr;k expected parking cost of a resident given parking
regime k with k¼pf for parking fees or k¼rp forresidential parking
pcnr;k expected parking cost of a visiting non-residentgiven parking regime k with k¼pf or k¼rp
ur overall utility a resident receives from living in thedistrict
unr overall utility a non-resident receives from visitingthe district
Ur aggregate utility of all residentsUnr aggregate utility of all non-residents (visiting and
non-visiting)Wk welfare given parking regime k with k¼pf or k¼rp
I. Molenda, G. Sieg / Economics of Transportation 2 (2013) 131–139138
fom lump-sum on-street parking fee (subsidy) in case
m¼ a; bflm lump-sum fee (subsidy) for parking in the lot in case
m¼ a; bα share of on-street parking spaces reserved for
residents
A.2. Proof of Proposition 2
To prove αnrαnr , where αn ¼ αn
r is only possible if αn ¼ 0 orαn ¼ 1, it is sufficient to show dWe=dαodUe
r=dα for αAð0;1Þ.Using solutions (33) for dWe=dα and (37) for dUe
r=dα, it holds thatdWe=dαodUe
r=dα if and only if
ved ~ue
nr
dα�dpcenr
dα
� �¼ ve � d ~ue
nr
dα�cnr � 1�α½ �ve'ðαÞþveðαÞþ1½ �
½1þveðαÞ�2� �
o0: ð50Þ
Recall from Lemma 1 that d ~uenr=dαo0 and dpcenr=dα40 hold in
our analysis. Accordingly, the sign of the term in brackets of (50)is negative. ve40 and thus dWe=dαodUe
r=dα holds for allαAð0;1Þ. □
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