The Economics of Parking:Road Congestion and Search for Parking

8/8/2019 The Economics of Parking:Road Congestion and Search for Parking

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The Economics of Parking: RoadCongestion and Search for Parking

Simon P. Andersony and Andr de Palmaz

June 2001, revised February 2002

A bst r act

Thi s paper examines t he benet s of pricing of parking for acent ral planner and for privat e operators. We t reat on st reetparking as a common property resource. Wi thout pricing, con-gest ion wil l be excessive close t o the desti nation, and wil l fallo more rapidly that is socially optimal . T he optimal pat tern is

at tained under privat e ownership if each owner contr ols parking(and prices in a monopolistically compet it ive manner) a gi ven

distance fr om the dest ination. I n t he case of congest ion bot h forparkers and on st reet (cruising for parking) t he case for privat eownership is less clear.

K E Y W O R D S: Parking, road congest ion, parking conges-ti on, common ressource.

We should l ike t o t hank Richard Arnott , St ef Proost and Robin Li ndsey for th eir

det ailed comment s. K iarash M ot amedi pr ovi ded research assist ance. The rst aut hor

gr at efull y acknowledges funding assistance from t he NSF under Gr ant SB R-9617784

and from the Bankard Fund at the Universit y of Virginia.y University of Vi rginia, Depart ment of Economics, 114 Rouss Hall, P.O. Box

4004182, Charlottesville, VA 22904-4182, USA.z Membre de lI nsti tu t Universit aire de France. Univer sit de Cer gy-Pontoise,

TH EM A , 33 Bd. du Port , 95100 Cergy-Pont oise, F RA NCE.

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1 I nt r od uct i on

Search for parki ng represents a maj or source of congest ion in urbanareas. Indeed, a signicant fract ion of t he trip t ime for in a congested

urban area is spent searching for a par king space. Arnott and Rowse(2001) report the claim that over half t he cars dri ving downtown in cit ieswith serious parking problems (l ike Bost on and major European ci t ies)

are cruising t o nd a parking space.M uch has been writ t en about the use of road pricing in ameli orat ing

congest ion problems. The formal economics lit erat ure is quit e developedin t hisarea and t he problem has received at tent ion from many prominent

economist s. Seminal work in this ar ea start s with Dupuit (1849), and

continues through Knight (1924), Boteux (1949), and Vi ckrey (1969).Road pri cing schemes are current ly used t o easecongest ion in Singapore,Hong K ong, and part s of the USA, Canada, Norway, and t he Nether-

lands, among ot her s.1 The t echnology for determi ning sophisticatedroad pricing tolls that depend on t ime of day and i nt ensit y of road usageis quite inexpensive. Yet t here remains signicant pol it ical resistancet o pricing of road usage in many jurisdicti ons, and consequently road

pricing remains scarcely used in t he urban context .Conversely, there is li t t le formal economic analysis of parki ng, al-

t hough technology for pricing parki ng is very simple (a parking meter!)

and there is li t t le social opprobrium for paying for parking. Arguably,ine cient search for parking may be at least as distort ionary as exces-sive road use. Clear ly opt imal policy should account for both sources ofcongest ion. In the absence of road pri cing, e cient pricing of parkingmay be an eect ive policy toll for combatt ing congest ion on the road

and in parking.Our object ive in t his paper is to study t he economics of parking by

sett ing up a simple and tractable model of parking congest ion. Previous

t heoret ical work on parki ng is scarce. Arnott and Rowse (2001) makea valiant att empt to model t he stochast icit y of the park ing process, but

t he model rapidly gets complex and this precludes them fr om gett ing t onormative issues. But, as Arnott and Rowse point out , t here remains

lit t le in the way of other formal l iterature, alt hough there is quite a lot ofdescript ive work, and some empirical work on mode choice that includeschoice of where t o park. Indeed, a search through the j ournals in jstort urns up no art icles wit h parking in t he t it le or the abstract!

We are interest ed her e primaril y i n parking that is unassigned, park-ing that drivers must search to nd. M ost car commuters havelong-termpark ing cont racts to avoid daily search for park ing. We are instead in-

1See t he discussion in Li ndsey and Verhoef (2002) for more elaborat ion.

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t erested mainly in the demand for parking from dri vers whose trips are

less regular, typicall y less predictable, and for shorter t ime periods. T henatural examples are shoppers, tourists, and deli very trucks, and weshal l t reat shoppers as our main example. We shall also treat on-street

park ing, so that t here is a xed and exogenous capacity at any locati on.On-street park ing lot s are either metered in pract ice, or else free. Privateownership is not usually viable, though we shall study that possibi lityas a theoret ical benchmark, and to also shed some light on the behavior

of the owners of (o-street) parki ng lot s.If parking is unpriced, then unassigned parking lots are a common

propert y resource that one would expect to be over-exploited in a free

entry equilibrium. Park ing space closest to the most desirable dest i-nat ion (t he CBD) are t he most coveted and will therefore be t he most overshed . Less desirable parking lots, further away, may end-up be-ing insu cient ly used because parkers overcrowd t he prime locations.

The opt imal pricing for parki ng charges for the congest ion externalit yand attens out the parking gradient by charging more at the parkingmeters for closer locations.

This l eads us t o i nvest igate how pri vate ownership of parking lots

in a market syst em can decentralize the opt imal conguration. We ndt hat private ownership of parking lots can do the job if private owner shipis diverse and monopol ist ical ly competit ive in t he following sense. Each

owner must insure that potential parkers nds his locat ion at least asat-

t ract ive as any other locat ion at equilibrium. This leaves t he owner witha trade-o between t he price charged by parki ng lot and the number ofdrivers who wish t o park t here. Owners t herefore face downward-slopingpark ing demands. These demands are higher for more desirable park-

ing locations: this contrasts wit h the standard Chamberlinian (1933)symmetry assumpt ion. While the parki ng lot owner is not a tradit ionalpr ice taker since he has lat itude in choosing his price, he is st il l akin t o

a competit ive agent because he faces an overall ut il it y constraint. Weshow t hat the social opt imum parking pattern is obtained under t his

market structure.Cruising for park ing slows down t hrough tra c. It does so by ne-

cessit at ing t ra c l ights at cross t horoughfares and by increasing t ra con the main highway because drivers circle to look for a vacant lot.Taking this feature into account introduces an ext ernali ty that is not in-t ernalized by parki ng lot owners. T his is because this externalit y is not

localized but rather i mpacts all t ravel lers passing through a locat ion. Int his case, private ownership does not render the market solution optimalbut t ends instead to creat e too lit t le parking at the prime locat ions andtoo much parking too far away.

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The organizat ion of the rest of the paper is follows. In Sect ion 2

we present the basic parking model assuming there is no externalit yimposed by t hose who cruise for par king on t hose who will park closert o t he CBD. We rst compare the equili bri um wit h unpriced parking

t o t he optimum and then we deri ve t he parking t ol l that delivers t heoptimum arrangement. We t hen show that this park ing fee is exact lyt hat chosen i n the equilibrium wit h private ownership. Hence, privateownership and opt imum are shown t o yield the same solution. In Sect ion

3 we introduce the externalit y from cruising and through tra c. Weestablish the manner in which the market solut ion diverges from theoptimum. Sect ion 4 provides a discussion of the pract icabilit y of the

private ownership solut ion. I t also discussed more general sett ing andt he abil it y of road pricing to att ain optimum outcome.

2 T he basic model

We are int erested in on-street parking that is not assigned to individualsin advance. Such is not the case for dail y commut ers, who ty picallysign long-t erm cont racts with private o-street lot operators. Shoppers,whose trips are less syst emat ic, t ypically do not have long t erm parking

contracts. Simi larly, tourists in the neighborhood of attract ions needt o park but do not reserve parking in advance. Deli very vehicles need

parking spots for l imited lengths of t ime and often at t imes that arenot easily predictable far in advance. It is such dr ivers who are mostly

likely t o be cruising for on-street parking. For concreteness, we shallt reat al l in t his category as shoppers. Most shoppers who drive comet oa shopping distri ct from a distant location and so we shall assume thatshoppers reside far away.

Formally, suppose that there is a common dest inat ion located atx = 0 (t hat we shall term t he CBD), and t here are N shoppers locatedfar away. The CBD i s at the end of a long, narrow city, and is served byparall el access roads. Perpendicular to theseaccess r oads are side-streets

t hat are used for on-street par king.2 Cars can park on street at any freelocation. Each shopper rst drives towards downtown (at speed vd),t hen starts to look for an empty parking spot on a side st reet according

t o a search process describ ed below. Final ly, once an available park ing

2Th e long-narrow city suer s from t he drawback t hat it is not symmet ric in t erms

of how drivers should search for parking spots along side streets: drivers arriving on

t he access roads at t he city edges wil l search inward from there. We do not t reat t his

asymmet ry i n what follows. A circular city (wit h radial access roads carryi ng t ra c

uniforml y fr om out side) avoids such asym met ry, and t he form al analysis would be

simil ar t o that below, apart from the addit ional complexi ty fr om t he two-dimensional

set -up. We have chosen t o present a long and narr ow city f or simpl icit y of exposit ion.

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spot is found, he walks (at speed vw) t o the CBD.

2.1 Sear ch for a par king spot

On-street parking is not assigned, so driver s must search for a vacant

park ing spot. The expected t ime to nd one at a dist ance x from theCBD depends on t he number of vacant and occupied parki ng spots att his locat ion. We assume that if a driver stops at locat ion x he willsearch at this location t il l he nds a vacant parking spot. The dri ver

t hen walks to the city. The walking cost is pr oport ional to x /vw . Thet otal number of parking places available on the interval [x; x + dx] fromt he CBD is denoted by K (x)dx, and for simplicity we set K (x) = k.

Hence, the city has width k; with t he CBD located at the end. T henumber of occupied parking places over [x; x + dx] is denoted by n(x)dxwith n (x) k.

We shall assume that the expected cost to an indivi dual who choosesto search for parking at location x is

E (x) = ck

k n (x): (1)

Here c is a scaling factor representing the money value of the t ime spentin searching for a vacant place. T he t ime spent is assumed propor-t ional to the ratio expression. This is increasing i n t he number of cars

parked, n (x), and at an increasing rate. I t goes to innity as thatnumber approaches the park ing capacit y, k. As well as exhibit ing theseintuit ive propert ies, t his closed form expression is convenient in whatfol lows to give a closed form t o the equili bri um and opt imum solut ions

of t he model.We shal l use (1) as the search cost for all indivi duals who choose

t o park at locat ion x. As we show the Appendix, t his funct ional formcan be given a search theoret ic microeconomic underpinning wit h drivers

choosing when t o search. The search cost for the last driver arriving atx follows a sequential search process wit h a cost c per lot inspected. Allpreceding drivers face lower on-sit e occupancy, and so lower search costs:t heir advantage is dissipated t hrough competit ion for ear lier posit ionsin

t he search procedure via timing of entry into the search process. Whatt his boil s down to is that the (inclusive) search cost at x is given by (1)for all individuals who choose to park at x (and the term inclusive -subsequent ly suppressed - refers t o the idea that the cost is t he act ual

cost of searching plus any costs due to sub-opt imally ear ly arri val).The form of theexpected cost of nding a park ing spot is purposefully

simplist ic. Arnot t and Rowse (2001) constr uct an elaborate microeco-nomic model of cruising for parking, and concludes that i t is intri nsically

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a very di cult pr oblem. By using a simple formulat ion t hat capt ures

t he key t rade-os, we are able to pursue the analysis of equili bri um andoptimum parking.

The expect ed cost from parking at locat ion x is:

C (x) =ck

k n (x)+ tx; (2)

where t is the net dollar cost per mile of walking instead of driv ing.3

2.2 Equi l ibr ium wit h unpriced park ing

In equilibrium, all parking locat ions must entail t he same expected cost,

and unused locations should ent ail a (weakly) higher cost . Denot e t hiscost by ec, and note that ec c since c is the minimum possible parkingcost borne by a drivers located right at the CBD and with one spot t o

check (which costs c). From (2), t he equil ibrium number of cars parkedat x solves:

ck[k n (x)]

+ tx = ec;

or:

n(x) = k

1 c

ec tx

; (3)

which must hold whenever n(x) 0. Let x denote t he fart hest distanceparked. T he car that is parked t he far thest away incurs the minimumsearch cost of c, so t hat x sat ises c + tx = ec; or:4

bx =ec c

t 0: (4)

We can now solve for the equilibrium expected cost by equating t hesupply and demand for park ing. This requires that

Z bx

0

n (x) dx = N . (5)

Integrating (5) with n(x) given by (3) gives:

k

bx ct

ln ec

ec tx

= N

3More precisely, t he t otal cost paid for a shopper whose point of ori gin is X is

E (x) + t d

X x

+ tw x . We can suppress t dX since it is constant. In (2) we have

also set t = t w t d wher e t w = w /vw and t d = c /vc . Here w is the value of

t ime for pedestri ans w hile c is t he value of t ime for drivers. Empir ically, we have

vw < vc and w > c so that t > 0.4Of course n ( bx) = 0.

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(where we not e that kbx is the maximum number of cars that could park

over t he int erval [0; x]). Substituting bx from (4) leads to:

N =kt

(ec c) + cln c

ec

: (6)

The equil ibrium cost ec(N ) is concave. Likewise, more shoppers extendt he parki ng boundary bx.

The expression ( 6) can also be writ t en as:

ec cln ec =tNk

+ c c ln c, (7)

which yi elds the equilibrium expected cost ec in implicit form. The RHSof t his expression is constant while t he LHS is increasing in ec in therelevant range where ec > c (which is required for bx > 0). The LHSis below the RHS for ec = c, and goes t o innit y wit h ec, so that therealways exists a unique solut ion for ec. T heother endogenous variables aret hen also uniquely det er mined. We defer a discussion of t he comparat ivestat ic propert ies of the unpr iced equi librium unt il we introduce cruising

congest ion in the next sect ion. The next step is t o deri ve t he opt imalparking pattern.

2.3 Social opti mum

Suppose the planner chooses t he optimal number of cars admissible ateach locat ion. However, once a car arrives at a parki ng locat ion, i t st i llmust search for a park ing place given t he search cost function E (x) de-scribed above. Clearly, t heoptimal allocati on of park ing involves parking

over an i nterval [0;xo], with n(x) > 0, for x < xo and n(xo) = 0. Thesocial planner faces t he foll owing problem of minimizing t he social costof gett ing t he shoppers to the CBD, or

SC =R x o0

c kk n(x)

+ tx

n (x) dx

s:t:R xo0 n (x) dx = N

The solut ion to the social problem involves marginal social cost (withrespect to n(x)) being equal for all locat ions with posit ive parking. Callt he mar ginal social cost , so that

ck2

[k n (x)]2+ tx = ; x 2 [0; xo]: (8)

Since n (xo) = 0, then

c + txo = : (9)

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This value can now be used in (8) to determine the optimal number

of shoppers parking at x as

no (x) = k

1

sc

c + t (xo x)

!

: (10)

The populat ion constraint can then be writ ten as:

Z xo

0k

1

sc

c + t (xo x)

!

dx = N :

Int egr at ing t he LHS, we get:

kt

2c + txo 2

pc

q(c + txo)

= N;

which implici t ly determinesxo and hencet heother endogenous variables.To nd an explicit expression for xo, it helps to wri t e this last equat ionusing t he value of in (9) as k /t

p

pc

2= N :

The optimized marginal social cost is therefore equal to:

=

0

@p

c +

sN tk

1

A

2

:

This tells us the optimal value of t he locat ion of the last parking places.

Subst it uting t his value of in equation (9) leads to:

xo =Nk

+ 2

sNk

r ct

: (11)

This expression is now to be compared wit h the (implicit) locat ion bxfrom the equi libr ium problem (see (4) and (7)).

2.4 Compar ison wit h t he unpr iced solut ion

The equilibri um parking arr angement has individual cost equalized at all

park ing locations, while t he opt imum arrangement has marginal socialcost equalized. The dierences in these arrangements are descri bed i nt he following proposit ion.

Proposition 1 T he parking span at the equili br ium with unpri ced park-ing is ti ghter than the optimal one. Moreover , there exists a location ~x< xo such that more cars are parked at x in t he equilibr ium than in theoptimum if and only if x < ~x:

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Proof. Recall from (3) that n(x) = k

1 cec tx

, while from (10),

no (x) = k

1 q cc+ t(x o x)

. We shall show that these schedules cross

once, at ~x. Indeed, there must be at least one crossing because bothschedules are continuous, wit h posit ive densit y at x = 0 and the integralof each of them over it s support is N . At any such crossing, ~x, n( ~x) =no ( ~x), and hence cec t~x =

qc

c+ t(xo ~x ) = (i.e., wehave cal led the common

value ). Denote derivat ives wit h pri mes. As wewill now show, [n( ~x)]0 ~x. Q.E.D.

The equilibrium involves tighter parking than is optimal because oft he uninternalized exter nali ty associat ed with parki ng. At equi libr ium,t here is more crowding, and hence excessive search cost, becausedriversdo not take int o account that select ing a parking spot close to t he CBD

increases t he search cost of a l arge number of other dri vers t rying t opark there. This is a variant of the classic common property resourceover-grazing or over-shing problem; see Gordon (1954) for a seminal

t reatment . Here, there is the extra twist since the resources ar e rankedby quali ty - spots closer to the CBD are int ri nsical ly more desirable.The analogy is that shermen sh t oo close t o the shore (see Weit zman(1974) for a similar idea). One might also surmise that at a common

propert y orchard, the lowest apples are over-picked.The optimum involves unequal treatment of equals in t he sense t hat

dierent individuals get dierent util it ies at the opt imum. Those who

are al located to park closer t o the CBD get higher util it y than thosewho park further away. The opt imum can be decentralized v ia pricingof parki ng. Since parki ng is more desirable closer to the cent er , theoptimum parking tari increases wit h closeness to t he CBD in ordert o counteract this eect and reduce t he over-congest ion t hat is most

pronounced closest to t he CBD. We next derive the opt imum parkingtari.

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2.5 O pt i m a l p ar k i n g t ar i s

We can use t he analysis above t o determine the optimum price of aparking lot as a funct ion of distance from the city. One can t hink of

t his as the rate paid at a parking meter. As we now show, the parkingmeter rat es that decentralize the optimum decrease with distance fromt he cit y center.

Theoptimal price (x) is equal to thedierencebetween themarginalsocial cost and t he private cost. Hence

(x) ="

ck2

[k no (x)]2+ tx

#

"ck

[k no (x) ]+ tx

#

; (12)

where no (x) is the opt imal number of cars parked at locat ion x. Rear-ranging (12) gives the optimal (p osit ive) price as:

(x) =c k no (x)

[k no (x)]2: (13)

The expression for the optimal parking densit y no (x) is given by (10)as:

no (x) = k

1

sc

c + t (xo x)

!

;

so the optimal parking t ari can also be writ ten (using (13)) as :

(x) = (c + txo) p

cq

c + t (xo x) + tx

; (14)

with xo given by (11).The rst term of this expression is t he marginal social cost of an

additional shopper parking at (any) x given the optimal occupancy (seeequat ion (9)). Hence, t he second t erm is the private cost (as can readily

be checked from equations (2) and (10)). The opt imal parking tari isa decreasing and concave funct ion of distance from t he CBD. I t is max-imum at x = 0, and zero at the fart hest parking place at xo. The shapeof the opt imal price schedule reect s the propert y t hat t he congest ionexternalit y diminishes wit h distance, and it does so at an increasing rate.

2.6 Par king lot oper at or s

We have just seen that the market equilibrium with unpri ced parking

lots is socially ine cient. This raises t he quest ion as to whether themarket mechanism can deli ver the opt imal allocation when prices aredetermined through market forces. The answer is a qualied a rmative,and we here explore the condit ions under which t his can be the case.

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We consider a market where park ing lots are managed by parking

lot operators. To approximate a competit ive market sett ing, we assumet hat the parking lots at x are pri ced by a single operator who set hisprice competit ively, i.e. by taken t he ot her prices as given. However,

each operator has a degree of market power since he controls all the lotsat x. This leads t o a sit uat ion t hat might be t ermed monopolist icallycompetit ive, since there are many operat ors and each has local mar ketpower. The set-up diers from the standard monopolist ic competit ion

framework because operators are not symmetric in t er ms of the mar ketcondit ions they face. Those located closer t o downtown have a compet-it ive advantage by dint of their more desirable locat ion.

Wi th parking lot operators, t he cost C (x) = E (x) + tx is augment edby a location specic price, p(x). T he equilibr ium condit ion is that t hisaugmented cost, denoted by Cm (x) for the monopolistic competit ioncase, is equal at all locations x at which there is parking, and is no lowerat any other location:

Cm (x) =ck

k nm (x)+ tx + p (x) :

To determine t he equil ibrium level of p(x), we consider the deci-sion problem faced by the individual park ing lot operator. The prot

of the operator located at x is (x) = p (x) nm

(x), with Cm

(x) =p(x) + C (x) = ~cm . Here t he expression C (x) is given by (2), wheren (x) is replaced by nm (x), the number of parkers given monopoli st i-cally competit ive pricing. The operator nonetheless faces a downward

sloping demand curve because a lower price wil l at tract more shoppers,whose presence raises the search cost. It is most convenient to subst ituteout for price and determine the operators choice of number of parkerst o attract. Doing so yields (x) = [~cm C (x)] nm (x). The rst-ordercondition is:

@(x)

@nm (x) = p (x) nm

(x)

"ck

[k nm (x)]2

#

= 0: (15)

Clearly, t he price charged by the operator is ther efore equal t o:

p (x) = nm (x)ck

[k nm (x) ]2: (16)

This shows:

Proposition 2 Private ownership of parking lots in a monopolisticallycompetitive market decentralizes the social optimum.

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Proof: This result is a direct consequence from t he fact that the

price p(x) charged by the parking lot operators (16), coincides wit h t heoptimal price (x) given by (13). Q.E.D.

We use t he fol lowing parameter values: c = 10 cent s, t = 4$=km,N = 20; 000 drivers, and k = 40; 000 park ing spots per kilometer:

1. The equili bri um cost is ec = $ 2:4186. The parking spans onbx = 0: 57965 km and average occupancy rate is 86: 259%. T hemaximum percent age of occupied parking space ( at x = 0) is95: 865%.

2. The opt imum cost (per driver) is SC/N = $ 1: 6963. Theoptimalpark ing span xo = 0: 72361km is larger than at theequilibrium andthe average occupancy rat e is smaller (69:098% ).

3. With monopolist ic operators, theuser cost ~cm = $ 2: 9944 is higherthan at equi libr ium; however, it is reduced t o $1: 6963 (t he valueof the social opt imum cost) if t he operator revenue is distributedas a lump sum. Note t hat the prot per parking operator is

smaller since some park ing place remain vacant; it is equal to:($ 2: 9944 $1: 696 3) 69:098% = $0:89696.

4. Finally, with the Int er net technology, the average user cost is $ 1(it is the minimum cost given ful l parking occupancy ).

3 Road congest ion from cr uising

We have assumed so far that the only congest ion eect is in t he cost of

searching for a parki ng lot. However, cars that are cruising and searchingfor a park ing place slow down other cars passing by. Cruising cars slowdown tra c direct ly on t he main arteries into town if they are searchingon t hose streets. They may also slow down tra c on the main art eries

if the cruisers are searching in side-streets: after an unsuccessful search,t he car searching for park ing must eit her r e-enter t he main stream orcross it to get to another side-st reet . The more such tra c there is,

t he greater the slow-down in terms of t ra c light delays, t ra c owinterference, etc.

Thus the speed at which through tra c can travel is reduced themore cars are looking for somewhere t o park . This is a second type of

externalit y germane to t he cruising pr oblem.

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3.1 A model of cr uising for par king

Without congest ion from cruising, the user cost is C (x) = E (x) + tx (see(2)). We now i ntroduce road congest ion by assuming t hat the driv ing

t ime taken to traver se the str et ch of road [x; x + x] is higher t he morepeople are searching for parking in this interval.

The simplest formulati on is to st ipulat e that a cruising car at x in-duces an extra delay for al l cars crossing [x; x + x]. We assume thatt he tot al delay induced by the cruising car at locat ion x is proport ionalt o t he number of dri ver s cruising for par king in this int erval, i.e. theadditional congestion is n (x) x, where is the ext ra delay cost percruiser (previously,= 0).5 This means that the expected cost incurred

by a shopper who par ks at x is:

C (x) =ck

k n (x)+ tx +

Z x p

xn (u) du; (17)

where xp is t he locat ion of t he car parked t he furt hest away.6 The thirdt erm in (16) is a non-localized ext ernali ty since drivers at x impact alldrivers park ing closer downtown. One advantage of our formulation ist hat aggregate congest ion is independent of the distribut ion of those

parking. To see t his, note t hat t he total cruising congest ion cost is

R x p0 n (x)

R x px n (u) dudx. Letting G(x) denote

R xpx n (u) du, we can writ e

t his cost as Rx p0 G0 (x) G(x)dx =

hG2

(x)2

i xp

0 = N , where t he laststep follows from the fact that G(0) = N and G(xp) = 0. Thus the totalcruising congest ion cost is constant, as claimed. This property impliest hat the solut ion t o the social problem is the sameas in sect ion 2, where

= 0.

3.2 Equi l ibr ium wit h unpriced parking and cruis-

in g congest i on

The equili bri um condition is that C (x) (given by (17)) is constant overall locations x at which there is parking and that i t is no lower at anyother locat ion. Since the last locat ion xp involves n (xp) = 0, the equi-

5Equivalent ly, tr avel t ime from x pt o x isR x p

x(1=v [n(u)]) du;where v [n(u)] is t he

speed given t hat t he densit y of cruisers i s measured by n (u) , with v0[:] < 0. We

assume here a simple inverse functi on, v [n( u) ] = 1/ n ( u) .6Just as t is i nterpreted as the net cost of walki ng over dri ving, so can be

int erpreted as t he net burden from cruising inter fer ence of d ri ving over walki ng.

This would allow for t o be negati ve if many t ra c light s slowed down pedest ri ans

more t han driver s. I f t he shopper get s a subway downtown aft er parki ng, then t here

is no int erference to pedestri ans, and is posit ive. For concr et eness we t reat posit ive

in t he sequel.

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libri um relat ion between the common equilibrium cost ~c and xp is:7

~c = c + txp: (18)

Clearly, the equilibrium involves a band of on-street parking cont iguoust o the CBD. Over this int erval, expected cost is constant, so that:

dC (x)dx

=c k n0 (x)

(k n (x))2+ t n (x) = 0, x 2 [0; xp] (19)

The solution of this equat ion is denot ed by ne (x); it is unique andexplicit . Note t hat t he number of parkers may ri sewit h distancefrom the

CBD, x, i f is largeenough: sincedr ivers wish t o avoid congest ion fromcruisers, they t end to park far away and t hen walk. This induces otherdrivers to act in a similar manner and as a consequence all dri vers endup parki ng far away. The induced excessive walking t ime is a source of

addit ional deadweight loss since locating far away is individually e cientbut col lect ively ine cient (t he total cruising t ime stays t he same). Int he sequel, we assume that k < t ; in the present case, this ensures that[ne(x)]0 < 0.8 We now turn t o the quali tat ive propert ies of t he soluti on.

3.3 Comparative stat ic propert ies of t he equi l ib-

rium solution

The equilibrium solut ion has the fol lowing comparative stat ic proper t ies.

Proposition 3 T he equili br ium parking span with unpri ced par king istighter when:

( a) the parking search cost, c; is lower;( b) the travel cost dierential, t ; is higher;( c) the crui sing congesti on cost, ; is lower.

Moreover, in each case, there exists a location ~xi ; i = a;b;c such thatmore cars are parked at x in the equilibri um before the change than afterthe change if and only i f x < ~x i :

Proof: We prove the statement for (a); the ot her statements areproved in a similar fashion. Let c0 < c1, and consider the correspondingschedul es n0 (x) and n1 (x). T hese must cross at least once becauseboth schedules are continuous, with positive density at x = 0 and theintegral of each of them over i ts support is N . At any such cr ossing, ~x,

7Similarly, at x = 0, ~c = c kk n (0)

+ N .8Since [ne(x) ]

0has the same sign asn (x ) t, theconditionk t < 0 is su cient,

but not necessary t o have [ne(x )]0

< 0.

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n0( ~x) = n1 ( ~x) n. A s wewil l now show, [n0( ~x) ]0 < [n1 ( ~x)]

0. From (19),

we have:

kc0(k n)2

[n0( ~x)]0

+ t n =kc1

(k n)2[n1( ~x)]

0+ t n = 0:

Therefore c0 [n0( ~x)]0 = c1 [n1( ~x)]

0and so [n0( ~x)]0 < [n1 ( ~x)]

0. This implies

t hat there is a unique crossing. The fact that the equili bri um densit yis sloping down mor e steeply at the crossing means that t he density forc0 is above t he one for c1for x < ~x and the converse is true for x > ~x.Q.E.D.

The int uit ion behind the result is as follows. A lower c means thatdrivers are less sensit ive t o parking congest ion. The new equilibriumt herefore involves more congested parking at each l ocation close to t hecit y and the total area devot ed t o parking fal ls. I f t rises, walking be-comes more of a nuisance relative t o driving, and parking lots closer inare more packed. If falls, t here is less annoyance from those cruisingfor parking, so shoppers will tend t o dri ve further in. Again, t his creates

more intense usage of parki ng lots furt her in.These results can be extended to det ermine the eect s on ot her pa-

rameters of interest. In part icular, we can nd out how equili bri um

t rip costs respond to lower search costs, c. Recall ing t hat ~c = c + txp(by (18)), then t he total cost incurred per driver falls by more than t hedecrease in the search cost, c, because xp also falls (average walki ng dis-t ance fall s). Hence if an improvement in park ing infor mation t echnologyallows dri vers to improve their search for parking, t hen the overall social

gains may be larger than t he simple reduct ion i n t he search cost.

3.4 Equil ibr ium with pr ivat e par king lot oper at or s

In t he absence of cruising for parking aect ing t ravel t ime, an operatorof a private parking lot has full control over the level of congest ion at xand therefore t he market out come can decentrali ze the social optimum(Proposit ion 2). By contrast, with the cruising ext ernal ity, there is anaddit ional cost that shopper s who cruise for parking impose on other

drivers who park at ot her locations (and t hereforewho are not generat ingpr ot at x). This second externalit y is not fully internalized by theparking operators since it is not a localized externality: t his cost aectsall t he cars which park downstream (nearer to t he CBD).

The equilibrium condition with parking lot operators is that C (x)asgiven by (17)) is augment ed by a location specic pri cep (x). LettingCm (x) again denote the cost in the monopolistic competit ion case, so

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Cm (x) = p(x) + ckk nm (x) + tx +

Z xmp

x nm (u) du: (20)

This cost is equal at al l locat ions x at which there is parking, and isno l ower at any other locat ion. Analogous t o our argument of the

previous sect ion, the prot of t he operator located at x is (x) =p(x) nm (x). Since Cm (x) = p (x) + C (x) = ~cm, we rewrit e this as(x) = [ ~cm C (x)] nm (x). The rst-order condit ion is t he same aswithout cruising congest ion, since the pri vate operator does not direct lyt ake into account the congest ion due to cruising for parki ng:

@(x)@nm (x) = p (x) nm (x)

"ck

[k nm (x)]2

#

= 0: (21)

Subst it uting into (20) and sett ing Cm (x) = ~cm gives the equilibriumcost level under monopolist icall y competit ive pricing as:

~cm = nm (x)

"ck

[k nm (x)]2

#

+c k

k nm (x)+ tx +

Z xmp

xnm (u) du:

The spat ial equili br ium wit h monopolistically competit ive pricingsatisesthe condit ion d~cm/dx = 0. Wri t ing this out gives:

" ck [nm (x)]0

[k nm (x) ]2+ t nm (x)

#+

ck [nm (x)]0

[k nm (x)]3[k + nm (x)] = 0: (22)

Again t he assumption k < t insures that [nm(x)]0 < 0. We can nowcompare this regime to the equili bri um one wit hout pricing:

Proposition 4 T he equi li br ium parking span without pri cing i s t ighterthan the monopolisti cally competi ti ve one. Moreover , there exists a loca-tion ~xm such that more cars are parked at x in the equili br ium than inthe monopoli st ic competi tion regime if and only if x < ~xm:

Proof: Theschedules ne (x) and nm (x) ar econtinuous, have posit ivedensity at x = 0; and the integral of each of them over it s supportis N . The therefore must cross at least once. At any crossing, ~xm ,ne( ~xm ) = nm ( ~xm ) n. From (22) and (19), we have:

ck(k n)2

n[ne( ~xm )]0 [nm ( ~xm )]0

o=

ck(k n)3

(k + n) [nm ( ~xm )]0:

Hence [ne( ~xm )]0 < [nm ( ~xm )]0 since the RHS is negat ive, and so there isa unique crossing. The equili brium densit y slopes down more steeply at

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t he crossing. Thus the equili bri um density is above the optimum one for

x < ~xm and conversely for x > ~xm . Q.E.D.

We have ~c = c + txp and ~cm = c + txmp . Since xmp > xp, the cost

paid by drivers is l arger when parki ng is priced monopolist ically com-petit ively, i .e., ~c < ~cm . Here, pricing increases user costs. However, if t her ms revenue were t o be lump-sum redistri but ed to dri ver s, cost would

decrease, since total search costs decrease. This means that moving to amarket system improves social surplus, but ther e is an adverse distri bu-t ional aect on drivers. The impact of pricing is dier ent in a dynamic

bott leneck model, la Vickrey (1969) and Arnott , de Palma, and Lind-sey (1993). These author s have shown t hat the impact of road pricingis t o shift congest ion from the users to the planner. In the bott leneckmodel, the user cost remains the same and t he social saving equals t he

decrease in t he amount of congest ion.9

3.5 Social opt im um wit h cr uising ext ernalit ies

As befor e, t he problem facing the social planner is to minimizethe social

cost SC =R xfp0 nf (x) C (x) dx or

SC =Z xfp

0nf (x)

"c k

k n f (x)+ tx +

Z xfp

xnf (u) du

#

dx

s:t:Z xfp

0nf (x) dx = N;

where the superscript f denot es the f irst-best solution and where C (x)is now given by (17)).

The optimum choice of the n (x), denoted by nf (x), satises:

="

ckk nf (x)

+ tx + Z xfp

xnf (u) du

#

+ nf

(x)

"ck

[k n f (x)]2

#

+

Z x

0 nf

(u) du;where is the common value of t he marginal social cost. Dierent iat ingt he RHS wit h respect to x tell us how the parking gradient behaves:

(ck [nf (x)]0

[k nf (x)]2+ t nf (x)

)

+

(ck [nf (x)]0

[k nf (x)]2+ t n f (x)

)

+

(ck[n f (x)]0

[k nf (x) ]3

k + n f (x)

)

+nnf (x)

o= 0: (23)

9With hypercongestion, even the user cost falls with road pricing.

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The optimum tari that decentralizes the social optimum is (x) =

C (x), or

(x) =c k nf (x)

[k nf (x)]2+

N Z x fp

xn f (u) du

!

:

Therefore the price schedule is the same as in the case = 0, butt he planner who r ealizes t hat the cruising for parking is dist ort ionary

when cars are parked e cient ly just subtract the cruising external-it y from i t pri ce so t hat the cruising externali ty washes out for theusers. Anot her similar more conventional) way to see it is t o not ice

that

N

R x fpx n

f

(u) du

=

R x0 n

f

(u) du represents the cruising ex-t ernalit y t hat a driver located at x imposes on al l the dr ivers l ocateddownstream.

We can now compare this regime wit h t he monopol ist ical ly compet-

it ive equilibrium.

Proposition 5 T he r st-best opti mal parking span is ti ghter than themonopolistically competitive one for > 0. Moreover, there exists alocation ~x f such that more cars are parked at x at the optimum than atthe monopolistically competitive regime if and only if x < ~x f :

Proof: Assume > 0. Foll owing t he argument of t he previouspr oposit ion, it su ces to show that:

hnf ( ~x f )

i 0

hnf

~x f

i 0, and the result

fol lows direct ly using the previous arguments. Clearly, if = 0, t he two

solut ions coincide. Q.E.D.

Proposit ion 4 showed t hat the monopolist icall y competit ive parking

gradient is atter t han that for the unpriced equili bri um. Proposit ion5 showed that the optimal gr adient is steeper than t he monopolist icallycompetit ive one. The reason for these result s can be understood frominspection of (23) which characterizes the social optimum, and which we

redisplay here:

(ck [nf (x)]0

[k nf (x)]2+ t nf (x)

)

+

(ck[nf (x)]0

[k nf (x)]3hk + n f (x)

i)

+nnf (x)

o= 0:

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The rst t er m in curly brackets is t he one that stems from t he unpri ced

equili bri um problem, and is ident ical to the condit ion (19) that char-acterizes equilibri um there. T he monopolist ically competit ive case addst hesecond term in curl y bracket s, so that therst twoterms are identical

t o the condit ion in (22) that characterizes equilibri um for monopolist iccompetit ion. This second term reects t he parki ng congest ion eectt hat is internalized under monopolist ic competit ion. Since this term isnegative, the slope is atter wit h monopolist ic competit ion. The nal

t erm reects the cruising externalit y no accounted for under monopolis-t ic competi t ion. Since t his term i s posit ive, t he gradient at the optimumis steeper than under monopolist ic competi t ion. The fact that the two

externalit ies play in dierent direct ions means that either the monopo-listically competitive or the unpriced equilibrium may be closer to theoptimum.

3.6 N umer ical example wit h cr uising

We use, as before, the following parameter values: c = 10 cents, t =4$=km , N = 20; 000 drivers, and k = 40; 000 par king spots per kilome-t er. We assume that = 10 4. For those values wit h computed thedierent solutions displayed in the gure below. Let y(x) = n (x) /k .

0

0.2

0.4

0.6

0.8

x

0.2 0.4 0.6 0.8 1y(x)

Parkers (horiziont al axis) versus location (vert ical axis)

This gure displays four curves:

1. Equilibrium ( > 0): so l i d t h ic k b l a c k c u r v e ;

2. Optimum ( 0 or monopolist ic competit ion= 0)): d i a m o n d ;

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3. Monopolist ic compet it ion ( > 0): c r o ss;

4. Equilibrium (= 0): t h i n so l i d l i n e

Not e t hat the average cruising cost is N2 = 9:09 cents. For theequilibrium cost wit hout cruising: ec = $ 2:419. Wit h cruising the equi-libri um cost raises to $2:68. Therefore, wit h cruising, the user cost raisesby 17:1 cents duet o t he redistribution of t he parkers and by 9 cents duet o cruising. The cost wit h private operators and cruising is $ 3:3. Withequal redistri bution of toll s, it is reduced to $ 3: 3 $0:658 = $2: 542(which is sl ight ly small er than the equil ibrium cost $2:68).

The major nding is that t hesituat ion with monopolist ic competit ion

leads now to about thesamecost than the sit uat ion without pricing (t hisdierence is 12 cents). T his is due to the fact that the pri vate operatorshave no control on cruising.

4 A lt er native mar ket st r uctur es and cost -sensit ive

demand for shopping

The monopolist ically competi t ive market st ructure enables the optimalsolut ion to be att ained when = 0, but it involves insu cient occupancyclose to the CBD when > 0. As wehave shown, t he monopoli st ic com-petit or int ernali zes the local external it y from search for parki ng at theoperator s locat ion, but does not take into account t he ext ernali ty fromcruising t hat hampers t ra c ows for those parking closer to t he CBD.

In that sense the operator sets a local price that is too low. The pricedistortion is greatest at location far away (t hrough which many driversmust pass) because the ext ernal it y is greatest at locations t he furt hestout. The pri ce gradient in monopolist ic competit ion is t oo shal low, and

t he optimal arrangement has more packed parking close to t he CBD.If instead al l parking wer e owned by a single monopoly rm, then

t his rm would internalize all of t he externali t ies, local as well as global.It would then reach the full socially opti mal ar rangement of parki ng.

However, monopol ies cause dist ort ions when demands are not perfectlyinelast ic. T his leads us t o consider the opt imalit y of t he al ternat ivearrangements when the demand for shopping is no longer xed at N .

Suppose instead that the number of shoppers dependsin a decreasing

fashion on t he full cost paid. Wit hout too much violence to the notat ion,let N ( ~c) be the number of shoppers as a funct ion of the full price, ~c,so this is the downward-sloping (aggregate) demand for shopping trips.

Consider rst the optimum problem. T he way we have set this up is t oderi ve the marginal social cost, , associated to an allocation. Since ist he marginal social cost of an extra shopper, then the opt imal number of

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shoppers is simply given by N ( ) from t he demand curve for shopping.

Asargued above, this optimum can be attained with appropriate parkingmeter fees (and t he same is true for > 0) so that all individuals payt he marginal social cost when t he opt imal fees (x) are in place. When = 0, we already know that the monopolist ically compet it ive priceexact ly t racks t he fee . This means that the ful l cost is the same andt he full social opt imum is att ained even wi th cost-sensit ive demand.

By contrast (st il l for the case = 0), the unpriced market equilib-rium i nvolves a lower cost level paid per shopper at the equi libr ium wefound with xed N . This t ranslates into too many shoppers when Nis vari able. At the other extreme, a monopolist of all parking lots will

use it s market power to raise t he pri ce too high, leading t o t oo l it t leshopping. This eect wil l st ill be present when is posit ive; so, despit et he proper ty that the monopolist will set the right allocat ion across lo-cations for any given level of N , i t wi ll price t oo high in aggregate anddeter shopping excessively.

Fi nally, one could consider int er mediat e degrees of market power be-tween monopolist ic competit ion and monopoly. Firms might be envis-aged as controlli ng each a tranche of locations. Al though we do not for-

mally consider this case, it seems straightfor ward t o conjecture that theresulting allocation lies between the monopoly and monopolistic cases,drawing the benets and disadvant ages from each. That is, prices tend

t o be too high from t he market power side, but all ocat ions are improved

t he larger the tr anche controll ed by a r m because t hen the cruisingexternalit y is int ernali zed t o a greater degree.

5 Conclusion

The economics of on-street parki ng pose a common-propert y resourceproblem. I f parki ng is unpri ced (or pri ced independently of locat ion)t hen t he market equilibrium will be t hat the resource is overused whereit is most valuable (at t he CBD) and the parking gradient will tail o

t oo fast. If t here is no pri cing at al l, too many shoppers will be attractedt o t he CBD. The social ly optimal conguration can be attained underpr ivate ownership of the parking lots. T his is not a great surprise from

t heCoaset heorem, although t he form of market structure (which wecant hink of as linked to t he way in which the privat e ownership must bestructur ed) needed to get the optimum is more intricate. First , agentsneed to be small enough ( i.e., own few enough parking lots) to act com-

petit ively by taking as given the ut i lit y level that can be attained bydrivers parking in others lots. Second, they need to be large enough t obe able to internalize the local externality in parking congest ion. Thismeans they must contr ol the lots at a given locat ion. Each par king lot

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operator therefore acts in a monopoli st icall y competit ive manner, tak-

ing as given the overall ut il it y constraint but wit h the power t o set pricegiven t hat occupancy will adjust. I f lot owners are larger, and have moremarket power, they will tend to set prices t oo high and so t here will be

t oo li t t le shopping act ivi ty. However, a full monopoly owner will fullyinternali ze the problem of cruising act ivit y that adver sely impacts t heroad speed of dr ivers parking elsewhere. The small-scale owners will notdo t his. This gives a t rade-o between e ciency across locat ions (that

t he full monopoly gets right) versus overall too high pri ces (t hat themonopoly encourages).

There are ot her reasons for not using a mar ket solut ion for on-street

park ing. One is t he transact ion cost involved if there are many small-scale operators - t hink for exampleof the enforcement problem of parkingwithout paying. Arguably too, local governments have wider objectivesin their park ing policy t hat simple e ciency. For example, many park-

ing meters have a (short ) t ime limit and regulat ions against coming backt o feed them. Even if someone - a commuter, say, who wants t o parkfrom dawn till dusk - is willing to pay more than the what individualshor t-term users would pay over the day, he is barred from doing so.

Arguably the local government is encouraging shopping at downtownestablishments; and subsidies are needed because of shopping externali-t ies and other implicit subsidies and market fail ures in other sectors.

However, Internet markets may provide a workable solut ion to the

common property problem. To x ideas, suppose t here were a plannerwho could costlessly al locate a specic parki ng spot to each shopper.Then there would be no search costs (apart from nding ones designatedplace!). Such a planner would allocatek cars to each locat ion. T his is thefull rst-best optimum, wit hout the search t echnology constraint . Thisr st-best optimal solut ion might be decent ralized in a market systemif property rights are properly dened and t ransacti on costs are zer o

(or, hopefull y, su ciently low). Dening propert y rights means havingowners for the park ing spaces, and, aswe have discussed, we need a large

number of owners for t his to be done e cient ly in the pri vate domain(alt hough onecould also imaginet hat public ownership can st ill b evi able

with an auction system).10 For market t ransact ions to be low-cost, weneed to have a uid spot market for parking lots. One might envisagesuch a market developing over the Internet, with parki ng lots auct ioned

10 We showed in Section 3 that th e monopolist ically competit ive (diverse owner ship)

solution diverges from the social optimum when > 0 (congest ion from crui sing).

However, i f parking l ots are perfectly assigned, t here would be no cruising. So t hent here would be no congesti on from cruising and t he diverse ownershi p equil ibri um

would coincide wi t h t he rst -best opt imum.

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o to t he highest bidder. There are some obvious benets from such a

system. Clearly el iminating cruising reduces aggregat e walking t ime andpracti cally el iminates search t ime. T he corresponding road congest ionand emissions decli ne t oo. On the supply side, land use for parking is

reduced and so are construct ion costs for park ing faci lit ies.The model can be usefully expanded in a vari ety of direct ions. Several

of t hese have to do wit h roads and road pricing. Road congest ion hasonly been addressed obli quely via the cruising cost externalit y. But

road congest ion also t ypically depends on the number of vehicles on t heroad at a part icular point (and downstream, in case of bott lenecks). Wehave assumed that the amount of parking space is exogenous. However,

freeing up park ing places eases on-road congest ion. Road pri cing shouldalso optimally be used in conjunct ion wit h parking pricing. This angleleadsus to consider how road pricing might be used if parking is unpri ced(or constant over locat ions). First, park ing ext ernali t ies can be aected

if the road price

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Pricing Systems on Network Performance: Transport ation ResearchA 34A, 407-436.

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for mation : A Stated Preference Approach: Transportation Re-search Booard, 71 st Annual Meeting, Washington, D.C.: PaperNo. 920280.

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paper 568.[28] Verhoef, E.P., P. Nijkamp and P. Ri etveld (1995),The Economics

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6 A p p endi x

One way of t hinking about a possible microeconomic foundation for t hisform is as follows. Suppose that each parking spot that a driver checks

t o determi ne whether or not i t is occupied incur s a doll ar search cost ofc.

Consider the last driver to arrive at locat ion x. T he probabilit yt hat a randomly sampled spot is free is given by p(x) = [k n (x)] /k .Suppose that search for a parking spot can be described by a stochast icpr ocess with replacement.11

We can now derive t he expect ed dollar cost E (x) of searching forand nding a vacant spot at location x. The probabilit y t hat the rstspot is free is p(x), and the associated cost is cp (x). Likewise, theprobabi lit y that the rst spot is occupied and the second spot is freeis (1 p(x)) p(x), and the associat ed cost is 2c (1 p (x)) p (x) becausetwo searches are involved. Therefore, t he expected dollar search cost is:

E(x) = chp(x) + 2[1 p(x)] p(x) + 3[1 p(x)]2 p(x) + :::

i:

Extracting p (x) from the series, we get E (x) = cp(x) d- /dp(x) ,where

- = [1 p(x) ] + [1 p (x)]2 ::: = [1 p(x)] /p (x) :

Therefore E (x) = c/p (x) = ck/ [k p (x)] as given i n t he t ext (see(1)).

This expected cost refers t o the last driver arriving at locat ion x.Drivers who arrive ear lier face lower search costs because fewer places

are already t aken. This means that there is a benet from early arri val:t he earlier the arrrival t ime, the lower the expected search cost. Sosuppose t hat drivers choose arrival t imes at x. Earl ier arrival entailsa cost in terms of sub-opt imally early arrival at the dest inat ion, but itcarr ies the benet of lower expect ed search costs. An equi librium i n

arrival terms at any x is such that each driver faces the same inclusivecost (early arrival cost plus search cost on arri val) and therefore fully

dissipates the rents due to an earlier posit ion in the search process.For mal ly, such an equilibri um can be deri ved as one in arrival t imes,whereby each arriving driver arrives at the t ime that just makes himindierent between arri ving then and t aking the corresponding place

in the search sequence, and arriving last of all (and incurring no early

11 Th is means th at an indivi dual who is cruising for parki ng forgets whet her she

has previously checked on a spot. W it h a large number of spots t o check, the expected

cost wit h r eplacement or wit hout r eplacement i s not very dier ent .

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arrival delay). Any driver who delays from arriving at the specied

t ime wil l nd her place in the sequence usurped by oneof the remainingdr ivers. This equil ibrium exhibits full dissipat ion of the rents accruing t obeing an early arri ver, as in Fudenberg and Tirole (1987) and Anderson

and Engers (1994). At this equilibri um, all of the dri vers face the samet otal inclusive search cost asthelast arr iver, whosecosts are given above.Thus the full social costs at x are simply n(x)C(x). In a similar vein,t he idea of schedule delay as a way to equalize congestion cost s across

drivers in equilibrium is also found in Arnott, de Palma, and Lindsey(1993).

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The Economics of Parking:Road Congestion and Search for Parking

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