Journal of Regional Science and Urban Economics tahun 2009
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REGIONAL SCIENCE AND URBAN ECONOMICS
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Regional Science and Urban Economics 39 (2009) 1–14
Contents lists available at ScienceDirect
Regional Science and Urban Economics
j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / r e g e c
Downtown parking in auto city☆
Richard Arnott a,⁎, John Rowse b
a
b
Department of Economics, University of California, Riverside, 4106, Sproul Hall, Riverside, CA 92521-0427, USA
Department of Economics, University of Calgary, Calgary, AB, Canada T2N 1N4
a r t i c l e
i n f o
Article history:
Received 12 December 2007
Received in revised form 11 July 2008
Accepted 12 August 2008
Available online 20 August 2008
JEL classification:
R40
Keywords:
Parking
Traffic congestion
Parking garages
Parking policy
a b s t r a c t
Arnott and Inci [Arnott, R. and Inci, E., 2006. An integrated model of downtown parking and traffic
congestion. Journal of Urban Economics 60, 418–442] developed an integrated model of curbside parking and
traffic congestion in a downtown area. Curbside parking is exogenously priced below its social opportunity
cost, and the stock of cars cruising for parking, which contributes to traffic congestion, adjusts to clear the
market for curbside parking spaces. Denser downtown areas have garage as well as curbside parking. Because
of economies of scale in garage construction, garages are discretely spaced. The friction of space confers
market power on parking garages. Spatial competition between parking garages, as modeled in Arnott
[Arnott, R., 2006. Spatial competition between downtown parking garages and downtown parking policy.
Transport Policy 13, 458–469], determines the equilibrium garage parking fee and spacing between parking
garages. Also, the stock of cars cruising for parking adjusts to equalize the full prices of curbside and garage
parking. This paper combines the ingredients of these two models, hence presenting an integrated model of
curbside parking, garage parking, and traffic congestion, and examines curbside parking policy in this context
through a numerical example with parameters representative of a medium-sized US city. The central result is
that raising the curbside parking fee appears to be a very attractive policy since it generates efficiency gains
that may be several times as large as the increased revenue raised.
© 2008 Elsevier B.V. All rights reserved.
1. Introduction
Anyone who has parked in the downtown area of a major city
during the business day will attest to its high cost. Parking in a
convenient parking garage is expensive, while finding cheaper
curbside parking normally entails cruising for parking and
walking some distance. To our knowledge, there are no reliable
estimates of the proportion of the average full price of a trip with
a downtown destination that is associated with parking. Informal
estimates of one half seem too high. It seems warranted to say,
however, that economists have paid less attention to downtown
parking than its importance merits. There is a large literature in
economics on urban auto congestion but only a few recent papers
on the economics of downtown parking (which will be reviewed
below).
☆ We would like to thank Eren Inci, Robin Lindsey, David Malueg, and seminar
participants at Clark University, Emory University, the University of California at
Riverside, the University of Colorado at Boulder, the University of Florida at Gainesville,
the University of Massachusetts at Amherst, and the University of California at Irvine for
helpful comments, and Junfu Zhang for pointing out an error in an earlier draft.
⁎ Corresponding author.
E-mail addresses: richard.arnott@ucr.edu (R. Arnott), rowse@ucalgary.ca (J. Rowse).
0166-0462/$ – see front matter © 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.regsciurbeco.2008.08.001
Arnott and Inci (2006) constructed an integrated model of curbside
parking and traffic congestion in an isotropic downtown area with
identical drivers and price-sensitive demand. The curbside meter rate
is set below its social opportunity cost. This results in excess demand
for curbside parking spaces. Parking is saturated, and cars cruise for
parking waiting for a parking spot to open up. The expected time spent
cruising for parking adjusts to clear the market, which is achieved via
adjustment in the density of cars cruising for parking. The cars
cruising for parking contribute to traffic congestion as well. Under
reasonable assumptions, Arnott and Inci demonstrated the existence
and uniqueness of steady-state equilibrium with saturated parking,
and also examined curbside parking policy in the context of the
model.
Denser downtown areas have garage as well as curbside parking.
Because of economies of scale in garage construction, garages are
discretely spaced. The friction of space then confers market power
on parking garages. Arnott (2006) developed a model of spatial
competition between parking garages, which generates an equilibrium parking fee that is above marginal cost. With underpriced
curbside parking and overpriced garage parking, the stock of cars
cruising for parking adjusts to equalize their full prices. This paper
combines the ingredients of these two models (except, to simplify, it
assumes inelastic demand for downtown parking), hence presenting
an integrated model of curbside parking, garage parking, and traffic
2
R. Arnott, J. Rowse / Regional Science and Urban Economics 39 (2009) 1–14
congestion, and examines curbside parking policy in this context
through a numerical example with parameters representative of a
medium-sized, auto-oriented city such as Winnipeg, Perth, San Diego,
Sacramento, or Phoenix.
The addition of garage parking alters the economics of downtown
parking in three interesting ways. First, the equilibrium condition
determining the stock of cars cruising for parking changes. With only
curbside parking, the stock of cars cruising for parking adjusts to clear
the market for trips. In contrast, with both curbside and garage
parking the stock of cars cruising for parking adjusts to equalize the
full prices of curbside and garage parking. Second, even though the
overpricing of garage parking does not create inefficiency directly,
since overall parking demand is assumed to be inelastic, it does so
indirectly in two ways, first, as noted already, by increasing the price
spread between curbside and garage parking and hence the stock of
cars cruising for parking, and second by causing parking garages to be
inefficiently small and too closely spaced. Third, the presence of
garage parking magnifies the distortion associated with the underpricing of curbside parking, or, put alternatively, increases the social
benefit of increasing the curbside parking fee. With only curbside
parking, the equilibrium full price of a downtown trip is determined
by the intersection of the trip demand curve and the curbside parking
capacity constraint. Raising the curbside meter rate does not alter this
full price of trips, but simply converts travel time (which includes intransit and cruising-for-parking time) costs dollar for dollar into meter
revenue, so that every extra dollar of revenue raised increases social
surplus by one dollar. But with garage parking, there is a magnification
effect. Raising the curbside meter rate does not alter the full price of
parking. Raising the curbside meter rate converts cruising-for-parking
time costs dollar for dollar into meter revenue. But there is the added
benefit that the reduction in the stock of cars cruising for parking
reduces traffic congestion, which benefits everyone. In our favored
numerical example, this magnification effect results in a $3.20
increase in social surplus for every dollar increase in meter revenue.
As noted above, the literature on the economics of parking is small.
We start by reviewing the broader literature, and then turn to the
small number of papers that distinguish between curbside and/or
garage parking or analyze cruising for parking.
Early work on the economics of parking argued that parking, like any
other commodity, should be priced at its social opportunity cost (Vickrey,
1954; Roth, 1965). Vickrey (1954) also developed a scheme for demandresponsive pricing of curbside parking. Over the next three decades,
parking was largely ignored by economists, in modal choice studies being
treated simply as a component of the fixed cost of a trip. Modern interest in
the economics of parking started in the early 1990s. Shoup (2005) has led
the way in generating interest in the economics of parking. In the 1990s, he
championed cashing out employer-provided parking, and has considered
many aspects of the economics of parking since then. Arnott et al. (1992)
and Anderson and de Palma (2004) extended the Vickrey bottleneck
model (1969) to analyze the temporospatial equilibrium of curbside
parking when all drivers have a common destination and desired arrival
time, such as for a special event or the morning commute. Arnott and
Rowse (1999) examined the steady-state equilibria of cars cruising for
parking on a circle when parking is unsaturated.
Arnott et al. (2005, Ch. 2, The basic model) presented a model that
examines the interaction between cruising for parking and traffic
congestion with only curbside parking. A more thorough treatment of
that model was provided in Arnott and Inci (2006). Several papers in
the literature have recognized that the stock of cars cruising for
parking adjusts to equalize the full prices of curbside and garage
parking (Calthrop, 2001; Shoup, 2005,1 2006; Arnott et al., 2005;
1
Shoup, Table 11-5, displays the results of 16 studies of cruising for parking in 11
cities over an eighty- year period. The mean share of traffic cruising was 30% and the
average search time was 8.1 min. While the study locations were not chosen randomly,
the results do indicate the potential importance of cruising for parking.
Calthrop and Proost, 2006). Calthrop (2001) and Arnott (2006)
considered the potential importance of garage market power,
Calthrop by assuming a monopoly supplier, Arnott by modeling
spatial competition between parking garages. The Los Angeles model
of Arnott, Rave, and Schöb includes curbside parking, garage parking,
endogenous cruising for parking, and garage market power, but
provides an unpersuasive treatment of garage market power. Arnott
(2006) contained all four elements as well, but focused on the
treatment of garage market power rather than providing a complete
analysis of the model. This paper provides a complete analysis
with the more satisfactory treatment of garage market power, and
also provides calibrated numerical analysis of a variety of parking
policies.
In terms of policy insights, our principal finding – which was
noted above – is that, under conditions of even moderate traffic
congestion, the social benefits from raising curbside parking rates
may be several times the additional meter revenue generated, a
double dividend result. Another important finding is that, with
realistic parameter values, less space should typically be allocated
to curbside parking the larger is the wedge between curbside and
garage parking rates.
Section 2 sets the stage by presenting a simplified model in which
garage parking is provided at constant unit cost. Section 3 presents
and analyzes the central model that takes into account the technology
of garage construction and spatial competition between parking
garages. Section 4 presents calibrated numerical examples for the
central model. Section 5 notes some directions for future research.
And Section 6 provides some concluding comments.
2. A simple model
Understanding the central model of the paper will be facilitated
by starting with a simplified variant. A broad-brush description is
followed by a precise statement.
2.1. Informal model description
The model describes the equilibrium of traffic flow and parking
in the downtown area of a major city.2 To simplify, it is assumed
that the downtown area is spatially homogeneous (isotropic) and
in steady state, and also that drivers are homogeneous. Drivers
enter the downtown area at an exogenous uniform rate per unit
area-time, and have destinations that are uniformly distributed
over it. Each driver travels a fixed distance over the downtown
streets to his destination. Once he reaches his destination, he
decides whether to park curbside or in a parking garage.3 Both
curbside and garage parking are provided continuously over space.
If he parks curbside, he may have to cruise for parking, circling the
block until a space opens up. After he has parked, he visits his
destination for a fixed period of time, and then exits the system.
Garage parking is assumed to be provided competitively by the
private sector at constant cost, with the city parking department
deciding on the curbside meter rate and the proportion of curbside
to allocate to parking. The curbside parking fee (the meter rate) is
less than the garage fee. Consequently, all drivers would like to
park curbside but the demand inflow is sufficiently high that this is
impossible. Curbside parking is saturated (the occupancy rate is
2
The model differs from that in Arnott and Inci (2006) in two respects. Arnott and
Inci consider the situation where all parking is curbside and the demand for trips is
sensitive to the full price of a trip. Here, in contrast, the demand for trips is completely
inelastic, and there is both curbside and garage parking. The model specification is
independent of the form of the street network, but for concreteness one may imagine
that there is a Manhattan network of one-way streets.
3
The paper does not consider parking lots. Parking lots are difficult to treat because
most are transitional land uses between the demolition of one building on a site and
the construction of the next.
R. Arnott, J. Rowse / Regional Science and Urban Economics 39 (2009) 1–14
100%) and the excess demand for curbside parking spaces is
rationed through cruising for parking. In particular, the stock of
cars cruising for parking adjusts such that the full price of curbside
parking, which is the sum of the meter payment and the cost of
time cruising for parking, equals the garage parking payment. The
downtown streets are congested by cars in transit and cruising for
parking. In particular, travel time per unit distance driven increases
with the density of traffic and the proportion of curbside allocated
to parking.
2.2. Formal model
Consider a spatially homogeneous downtown area to which the
demand for travel per unit area-time is constant, D. Each driver travels
a distance δ over the downtown streets to his destination, parks there
for a period of time λ, and then exits.4 A driver has a choice between
parking curbside, where the meter rate is f per unit time, and parking
in a parking garage, at a rate c per unit time, equal to the resource
cost of providing a garage parking space.5 Both curbside and garage
parking are continuously provided over space. By assumption (its
rationale will be given later), f b c, and the excess demand for curbside
parking is rationed through cruising for parking. The stock of curbside
parking is P per unit area, so that the number of garage parking spaces
per unit area needed to accommodate the exogenous demand is
Dλ − P. The technology of traffic congestion is described by the function
t = t(T, C, P), where t is travel time per unit distance, T the stock of cars
in-transit per unit area, and C the stock of cars cruising for parking per
unit area.6 The larger are C and T, the higher the density of traffic on
the city streets, so that tT and tC (subscripts denote partial derivatives)
are positive, and the larger is P, the lower the proportion of street
space available for traffic, so that tP is positive too. It is assumed as well
that t is a convex function of T, C, and P.
D is sufficiently high that, even if all curbside is allocated to parking
(so that P =Pmax), there is still a need for garage parking (i.e., Dλ N Pmax).
Due to the underpricing of curbside parking, the stock of cars cruising
for parking adjusts such that the full price of curbside parking, the sum
of the meter payment and the value of time lost cruising for parking,
equals the full price of garage parking.7 For the moment, it is assumed
that, even when curbside parking is provided free and all curbside is
allocated to parking, the street system can still accommodate the
exogenous demand.8
The density of cars per unit area is T + C, their velocity, v, is 1/t, and
since flow equals density times velocity, the flow in terms of cardistance per unit area-time is (T + C)/t. If there are M distance units of
4
One could equally well assume that each driver travels a distance δ/2 over the
downtown streets to his destination, parks there for a period of time λ, then drives an
equal distance over the streets to his entry point.
5
The paper neglects the subsidization of garage parking, which is practically very
important (Small and Verhoef, 2007, p.113). The bulk of employers, though a smaller
proportion of downtown employers, heavily subsidize their employees' garage
parking. Furthermore, many shoppers receive subsidized garage parking via “parking
validation” whereby a retailer pays for the garage parking of clients who have
purchased goods from his store. The subsidization of garage parking reduces the full
price of garage parking, hence the price differential between curbside and garage
parking, and hence the incentive to cruise for parking.
6
Realistically, travel time per unit distance is affected by cars entering and exiting
curbside parking spaces as well. This could be incorporated into the analysis by adding
the curbside parking turnover rate as an argument to the function t. The specification
of the technology assumes that P is divisible. This is not completely satisfactory since
having only one curbside parking space per block would slow down traffic almost as
much, and perhaps even more, as having an entire curb allocated to parking. This
objection can be accommodated by having a certain proportion of curbside blocks
completely allocated to curbside parking with the remainder being free of curbside
parking. But this would violate the spatial symmetry assumed in the model.
7
The paper ignores possible safety differences between curbside and garage
parking, as well as the search time and walking time inside the parking garage.
8
Primitive conditions for this assumption to hold are given in Section 2.6, which
examines the congestion technology in detail.
3
one-way streets per unit area, then a person standing on a sidewalk
would observe a flow of (T + C)/(Mt) cars per unit time.9 Throughput is
defined analogously to flow but includes only cars in transit10; thus,
the throughput in terms of car-distance per unit area-time is T/t.
Steady-state equilibrium is described by two conditions. The first,
the steady-state equilibrium condition, is that the input rate into the intransit pool, D, equals the output rate, which equals the stock of cars in
the in-transit pool divided by the length of time each car stays in the
pool, T/(δt(T, C, P)):
D¼
T
:
δt ðT; C; P Þ
ð1Þ
This may be written alternatively as Dδ = T/t(T, C, P). Dδ is the input
in terms of car-distance per unit area-time, and T/t(T, C, P) is the
throughput. Let ρ be the value of time.11 The second equilibrium
condition, the parking equilibrium condition, is that the stock of cars
cruising for parking adjusts to equilibrate the full prices of garage and
curbside parking:
cλ ¼ f λ þ
ρCλ
:
P
ð2Þ
The full price of garage parking is cλ. The full price of curbside
parking is f λ plus the (expected) cost of cruising for parking. The
expected time cruising for parking12 equals the stock of cars cruising
for parking, C, divided by the rate at which curbside parking spots
are vacated, P/λ. Thus, holding fixed the expected time cruising for
parking, the stock of cars cruising for parking increases with the
number of curbside parking spaces available. The cost of cruising for
parking equals the expected time cruising for parking times the value
of time. Eq. (2) may be rewritten as
C¼
ðc−f ÞP
;
ρ
ð3Þ
indicating the equilibrium stock of cars cruising for parking as a
function of c, f, P, and ρ.
This simple model has two equations in two unknowns, T and C.
The equations are recursive. Eq. (3) determines C and then Eq. (1)
determines T. Resource costs per unit area-time, RC, are simply ρ(T + C),
the stock of cars in transit and cruising for parking, times the value of
time, plus the resource cost of garage parking, c(Dλ − P).
9
Flow equals density times velocity is known as the Fundamental Identity of Traffic
Flow. Applying this identity in this context requires some care. Ordinarily, density is
measured per unit distance, so that, with velocity measured as distance per unit time,
the dimension of flow is cars per unit time. Here, however, density is measured as cars
per unit area, so that application of the formula gives flow in units of car-distance per
unit area-time. With M miles of city street per unit area, the density in terms of cars
per unit distance is (T + C)/M, and application of the Identity gives flow as cars per unit
time on a street, (T + C)/(Mt).
10
Since the transportation engineering literature has not analyzed situations in
which cars circle the block, it does not make a terminological distinction between flow
and throughput. It seems intuitive to define flow as the number of cars that a
bystander would count passing by per unit time. Throughput too seems an appropriate
choice of term.
11
The large empirical literature on the value of travel time finds it to differ
significantly across travel activities and for in-transit travel to differ according to the
level of congestion. The analysis could be extended straightforwardly to allow for these
considerations.
12
This statement takes into account that the service discipline is random access.
There are several features of the model that can be interpreted as either stochastic or
deterministic. A deterministic interpretation, with FIFO access to curbside parking on
each block, is the most straightforward. Stochastic interpretations are more realistic
but would need to be solved using queuing theory (see, for example, Breuer and Baum,
2005).
4
R. Arnott, J. Rowse / Regional Science and Urban Economics 39 (2009) 1–14
ρ(T + C) + c(Dλ − P). Thus, the constrained social welfare optimization
problem is to choose T, C, and P to
2.3. Full social optimum
The full social optimum entails no cruising for parking. The density
of in-transit traffic is then determined by Eq. (1) with C = 0. The
social welfare optimization problem is to choose T and P to minimize resource costs per unit area-time, subject to the steady-state
equilibrium condition, given by Eq. (1):
min RC ¼ ρT þ cðDλ−P Þ s:t: T−δt ðT; 0; P ÞD ¼ 0:
T;P
ð4Þ
The shadow cost of curbside parking is the increase in in-transit
travel time cost per unit area-time from having one more curbside
parking spot, which, from Eq. (1), is ρdT/dP = ρδtPD/(1 − δtTD)). If, with
P = 0, the shadow cost of curbside parking exceeds c, it is optimal to
allocate no curbside to parking. And if, with P = Pmax, the shadow cost
of curbside parking falls short of c, it is optimal to allocate all curbside
to parking. Otherwise, the optimal P solves
ρδtP D
−c ¼ 0;
1−δtT D
T;C;P
s:t:
iiÞ
Dδt ðT; C; P Þ−T ¼ 0; /
P
ðc−f Þ−C ¼ 0;
’
ρ
ð6Þ
where ϕ is the Lagrange multiplier on constraint i) and φ that on
constraint ii). The second-best optimum may entail no curbside
allocated to parking, in which case there is no cruising for parking,
or all the curbside allocated to parking. An interior optimum is characterized by the first-order conditions:
T : ρ þ /ðDδtT −1Þ ¼ 0
ð7aÞ
C : ρ þ /DδtC −’ ¼ 0
ð7bÞ
P : −c þ /DδtP −
’ðc−f Þ
¼0
ρ
ð7cÞ
Substituting out the Lagrange multipliers yields
ð5Þ
the level of curbside parking should be chosen to equalize the shadow
costs of curbside and garage parking.13
Let ⁎ denote the value of a variable at the social optimum. The
social optimum can be decentralized by setting P = P ⁎, T = T ⁎, and
f = f ⁎ = c.
2.4. Constrained (second-best) social optimum
Since parking policy is local, it is poorly documented and has been
little studied. One empirical regularity stands out, however, at least for
US cities. In downtown areas, the curbside parking fee is considerably
lower than garage parking fees. In downtown Boston, for example, the
meter rate has remained at $1.00 per hour for twenty years, while garage
parking fees are as high as $10.00 for the first hour. What accounts for
this price differential? We have posed this question to several seminar
audiences. The common answer is that downtown merchants' associations lobby City Hall to set the meter rate low in order to draw shoppers
away from suburban shopping centers, where most parking is provided
free. In many downtown shopping areas, merchants pay for shoppers'
garage parking by validating their garage parking stubs. Since most
curbside parking has time limits, and since merchants cannot pay for
their customers' curbside parking, a low meter rate subsidizes the
parking of short-term shoppers. If most curbside parking is occupied by
shoppers, the result is a form of price discrimination. Shoppers end up
paying less for downtown parking than non-shoppers. In this paper, we
do not attempt to model the political economy of downtown parking
policy, but instead simply assume that that the meter rate is set low –
below the marginal cost of a garage parking space – by the local
transportation authority, and explore the implications of the price
wedge between curbside and garage parking.
The constrained social optimum is now considered, where the
constraint is that the curbside parking fee is set below c, with the
stock of cars cruising for parking adjusting so as to satisfy the parking
equilibrium condition. The second-best optimal allocation of curbside to parking minimizes resource costs per unit area, subject to both
the steady-state equilibrium condition, (1), and the parking equilibrium condition, (2). Resource costs per unit area-time are given by
13
iÞ
min RC ¼ ρðT þ C Þ þ cðDλ−P Þ
Recall that the constraint is that the inflow to the in-transit pool equal the outflow.
In conventional traffic flow theory, there are two densities corresponding to a level of
flow. The specification of the minimization problems implies the choice of the lower
density. This complication is addressed in Section 2.5. The convexity of the congestion
function ensures that there is a unique minimum corresponding to the lower density.
ρDδtP
DδtC
þ ðc−f Þ þ
ðc−f Þ−c ¼ 0:
1−DδtT
1−DδtT
ð8Þ
A heuristic derivation is as follows: P should be chosen such that
dRC/dP = 0. From the objective function, dRC/dP = ρdT/dP + ρdC/dP − c;
from constraint ii), dC/dP = (c − f)/ρ; and from constraint i), (dT/dP)
(1 − DδtT) = DδtCdC/dP + DδtP. The last term in Eq. (8) is the marginal
social benefit of P, the reduction in garage costs. The other three
terms are components of the marginal cost, all of which relate to
travel costs. The first term captures the capacity reduction effect; this
is the increase in aggregate in-transit travel costs that comes about
through the reduction in the road space available, holding constant
T and C. The second term captures the cruising-for-parking stock
effect; since the stock of cars cruising for parking is proportional to the
amount of curbside parking, the increase in P increases the stock
of cars cruising for parking, which, holding T and P fixed, increases
congestion. The third term captures the cruising-for-parking congestion effect; via the steady-state equilibrium condition, the increase in
the stock of cars cruising for parking causes the stock of cars in transit
to increase, which further augments congestion.
Let ⁎⁎ denote values at the constrained social optimum. With the
curbside meter rate set at the exogenous level, the constrained social
optimum can be decentralized by setting T = T ⁎⁎, C = C ⁎⁎, and P = P ⁎⁎.
Unless both allocations entail the same corner solution, the optimal
amount of curbside to allocate to parking is greater in the full social
optimum than in the second-best social optimum with underpriced
curbside parking, i.e. P ⁎ N P ⁎⁎. In both allocations, the marginal social
benefit of increasing P by one unit is the saving in garage resource costs,
c. But the marginal social cost of increasing P is lower in the full social
optimum than in the second-best optimum since the costs deriving from
the cruising-for-parking stock and congestion effects are absent.
2.5. Revenue multiplier: the effects of raising the curbside parking fee
A principal theme of the paper is that the underpricing of curbside
parking is wasteful. To formalize this point, this subsection examines
the resource savings from increasing the curbside parking fee by a
small amount when it is below c, holding P fixed. From the expression
for resource costs:
dRC
dT
dC
¼ρ
þρ
df
df
df
dT
dC
¼ρ
þ1
:
dC ð1Þ
df
½ j
ð9Þ
where dT/dC|(1) denotes the change in T associated with a unit
change in C when Eq. (1) is satisfied. Now, the revenue raised from
R. Arnott, J. Rowse / Regional Science and Urban Economics 39 (2009) 1–14
5
the parking fee, R, is Pf, so that dR/df = P. From Eq. (2), ρdC/df = − P.
Thus,
−
½ j
dRC
dT
¼
df
dC
þ1
ð1Þ
dRdf :
ð10Þ
Hence, the resource cost saving per unit area-time from raising the
curbside parking fee equals a multiple of the increase in the parking fee
revenue raised. We term this the multiplication effect. Since the full price
of parking is c, whether a driver parks curbside or in a parking garage,
cruising-for-parking costs fall by exactly the amount of the increase in
parking fee revenue, and there is the added benefit that in-transit travel
costs fall due to the reduction in the stock of cars cruising for parking.
Define
μu−
dRC
dT
df
j
¼1þ
dR
dC ð1Þ
df
ð11Þ
to be the (marginal) revenue multiplier. What determines the size of
the revenue multiplier? Or put alternatively, by how much does a unit
reduction in the stock of cars cruising for parking reduce the total
stock of cars on the road? The answer depends on the technology of
congestion, as well as its level. The revenue multiplier is even larger if
account is taken of the marginal cost of public funds exceeding 1.
2.6. The congestion technology
The steady-state equilibrium condition, (1), can be written
implicitly as C = C(T; P, D). Holding fixed P and D, for each level of T
the function gives the stock of cars cruising for parking consistent with
steady-state equilibrium. Under the assumption that the function t(.)
is convex in T, C, and P, holding P fixed the function C is concave in T.
In the absence of cruising for parking, with realistic congestion
functions there are normally two densities consistent with a given
level of (feasible) flow, i.e. T = δt(T, 0, P)D has two solutions, one
corresponding to regular traffic flow, the other to traffic jam
conditions. Fig. 1 displays the graph of the function C, termed the
steady-state locus, with these two properties. An increase in P causes
the locus to shift down; holding T fixed, Eq. (1) determines an
equilibrium travel time, and to offset the increase in travel time due to
the increase in P requires a decrease in C. An increase in D causes the
locus to shift down; holding P and T fixed, for Eq. (1) to continue to be
satisfied the increase in D must be offset by a decrease in t, and hence
a decrease in C. If D increases sufficiently, there is no (T, C) satisfying
Eq. (1) and a steady-state equilibrium does not exist.
It is assumed furthermore that the function t(.) is weakly separable,
specifically that t = t(T, C, P) = t(V (T, C), P), with V defined as the effective density of cars on the road14 and normalized in terms of in-transit
car-equivalents in the absence of cruising for parking, i.e. V (T, 0) = T.
Define κ(C, P) = maxT T/t(V (T, C), P) to be the throughput capacity of
the street system as a function of C and P, so that κ(0, P) is capacity as
conventionally defined. It indicates the maximum entry rate, in terms
14
In the numerical examples presented later, it will be assumed that congestion
takes the form of a negative linear relationship between velocity and effective density:
v = vf (1 − V/Vj), where vf is free-flow speed and Vj is effective jam density, which is a
slight generalization of Greenshield's Relation. Travel time per unit distance is the
reciprocal of velocity. Then letting t0 denote free-flow travel time, this relationship can
be rewritten as t = t0/(1 − V/Vj). It will also be assumed that a car cruising for parking
creates 1.5 times as much congestion as a car in transit, i.e. V = T + 1.5C. Eq. (1) then
becomes T(Vj − T − 1.5C) = δDt0Vj. Thus, capacity is Vj/(4t0), the effective density
corresponding to this capacity is Vj/2, (dC/dT)(1) = (Vj − 2T − 1.5C)/(1.5T), and the revenue
multiplier is µ = (Vj − 0.5T − 1.5C)/(Vj −2T − 1.5C). We shall assume furthermore that
effective jam density is linearly decreasing in the proportion of curbside allocated to
parking: Vj = Ω(1 − P/Pmax), where Ω is the effective jam density with no curbside
parking.
Fig. 1. Steady-state equilibrium. Notes: 1. C(T;P,D) is the steady-state locus, Eq. (1).
2. C ¼ c−f
ρ P is the parking equilibrium locus, Eq. (2).
of car-distance per unit area-time, the street system can accommodate
in steady-state equilibrium for a given C and P. If Dδ exceeds κ(0, P),
then the entry rate exceeds throughput capacity even in the absence
of cruising for parking, and no steady-state equilibrium exists. If Dδ is
less than κ(0, P), then the steady-state locus lies in the positive
quadrant. The parking equilibrium condition, Eq. (2), can be written as
C = (c − f)P/ρ, giving the equilibrium stock of cars cruising for parking.
Since the condition is independent of T, its graph in Fig. 1, the parking
equilibrium locus, is a horizontal line. If Eq. (2) lies everywhere above
Eq. (1), which occurs if Dδ N κ((c − f) P/ρ, P), the entry rate exceeds
throughput capacity at the equilibrium level of cruising for parking,
and no equilibrium exists.15 If Eqs. (1) and (2) intersect, they do so
twice. A straightforward stability argument16 can be applied to
establish that the left-hand intersection point is stable while the
right-hand one is unstable. Thus, we take the left-hand intersection
point to be the equilibrium.
An initial equilibrium is indicated by E1 in Fig. 1. The revenue
multiplier equals one plus the reciprocal of the slope of the steadystate locus at the equilibrium point. If the curbside parking fee is
lowered, the parking equilibrium locus shifts up, causing the equilibrium to move up along the steady-state locus. Due to the convexity
of the congestion technology, the slope of the steady-state locus at the
equilibrium point falls, and hence the revenue multiplier increases.
Now consider the effect of increasing the amount of curbside allocated
to parking. The steady-state locus shifts down and the parking
equilibrium locus shifts up. The equilibrium T and C increase; effective
density increases, which, along with the decrease in road capacity,
causes traffic congestion to worsen; and the revenue multiplier can be
shown to increase.17
2.7. Complications caused by garage construction technology
The above analysis laid out the economics of equilibrium when
there is curbside and garage parking, when garage parking is priced at
15
Earlier, to avoid considering non-existence of a solution to the resource cost
minimization problems, it was assumed an equilibrium exists even when traffic is as
congested as possible, which occurs when f = 0 and P = Pmax. This condition is that Dδ b
κ(cPmax/ρ, Pmax). With a positive parking fee and/or less curbside allocated to parking, a
solution to the resource cost minimization problems (and the corresponding
equilibria) may exist when this condition is not satisfied.
16
It is reasonable to assume that the parking equilibrium condition is always satisfied,
so that trajectories travel along the Ċ = 0 locus. Ṫ is positive above Ṫ = 0 and negative
below it. If therefore the initial level of T lies below the Ṫ = 0 locus, T decreases, while if it
lies above the Ṫ = 0 locus, T increases. It follows that E1 is a stable equilibrium and E2 an
unstable equilibrium.
^
17
Substituting Eq. (2) into Eq. (1) gives T = δt(T, (c − f)P/ρ, P)D = T (P). Also, from Eq. (3),
^
^
C = (c − f)P/ρ. Then, dT/dC = (dT/dP) ÷ (dC/dP) = [ρ/(c − f)]dT/dP, and so d(dT/dC)/dP = [ρ/(c
^
− f)]d2T/dP2, which can be shown to be positive.
6
R. Arnott, J. Rowse / Regional Science and Urban Economics 39 (2009) 1–14
constant unit cost, and when curbside parking is priced below this
level. Unfortunately, the model is unrealistically simple in assuming
that garage spaces are supplied uniformly over space at constant unit
cost. The technology of garage construction and other factors result in
parking garages being discretely spaced.18 To reduce his walking costs,
a driver is willing to pay a premium to park in the parking garage
closest to his destination. Parking garages therefore have market
power and may exercise it by pricing above marginal cost. Furthermore, spatial competition between parking garages may result in their
being inefficiently spaced. Taking these considerations into account
complicates the economics, since there will then be three distortions
that need to be taken into account, not only the underpricing of
curbside parking but also the overpricing and inefficient spacing of
garage parking.
In the next section, the model of this section is extended to take
into account the exercise of market power caused by the discrete spacing
of parking garages. The exact reason for the discrete spacing of parking
garages is secondary. It is assumed that the discrete spacing arises
from the fixed land area required for a central ramp, which generates
horizontal economies of scale. The optimal spacing minimizes overall
resources costs. The equilibrium spacing is the outcome of spatial
competition between parking garages.
individual curbside parker to cruise for parking on his destination
block, or the spatial homogeneity of traffic flow.21 These assumptions
together imply that the steady-state equilibrium condition for the
simple model, T = δt(T, 0, P)D, continues to hold. Denoting the corresponding equilibrium in-transit density as a function of P by T ⁎(P)
gives TT = ρT⁎ (P).
Efficiency entails identical parking garages being symmetrically
arrayed over space, with diamond-shaped market areas. Let s be the grid
or Manhattan distance between parking garages, x the capacity of each
parking garage, and K(x) the minimum cost per unit time of a garage as a
function of capacity. Each garage services an area of s2/2. With demand
inflow D per unit area-time and parking duration λ, the total number
of parking spaces in a garage's service area is Dλs2/2. Since Ps2/2
curbside parking spaces are provided in the service area, garage capacity
is x = (Dλ −P)s2/2 and GC =K((Dλ −P)s2/2) ÷s2/2. Since the demand for
garage parking is uniformly distributed over space, the average distance
walked by a garage parker is 2s/3 so that average walking time is 2s/(3w),
where w is walking speed, and WC = 2ρs(D −P/λ)/(3w). Combining the
above results gives
RC ¼
2
K ðDλ−P Þ s2
s2
2
þ 2ρs
D− λP
þ ρT⁎ðP Þ:
3w
ð13Þ
3. The central model
The primitives of the model differ from those of the simple model
of the previous section in three respects. First, the garage cost function
incorporates horizontal economies of scale, reflecting the fixed costs
associated with the central ramp. Second, to avoid dealing with price
discrimination based on parking duration, parking duration rather
than visit duration is taken to be exogenous. And third, a grid street
network is assumed.
In many cities, there are both public and private parking garages.
To keep the analysis manageable, however, it is assumed that all
parking garages are private.19 The social optimum is solved first, then
the spatial competition equilibrium is solved when the government
intervenes only through its curbside parking policy.20
3.1. Social optimum
Since travel demand is perfectly inelastic, the social optimum
entails minimizing resource costs per unit area-time. There are three
components to resource costs per unit area-time: garage costs per unit
area-time (GC), walking costs per unit area-time (WC), and in-transit
travel costs per unit area-time (TT):
RC ¼ GC þ WC þ TT:
ð12Þ
It is assumed that the presence of parking garages does not alter
the distance drivers travel over city streets, or the decision of the
18
Suppose, for the sake of argument, that parking garages are continuously
distributed over space. It would be cheapest to construct garage parking on the
ground floor of every building, but this space is especially valuable for retail purposes.
Constructing below-ground parking may be cost effective at the time the building is
constructed, but is expensive for buildings that were originally constructed without
underground parking. Constructing above-ground parking in multi-use buildings
raises structural issues. In most situations, the cost of constructing garage space is
minimized with structures specifically designed as parking garages. Even if parking
were distributed continuously over space, parking garage entrances would not be.
12/4/08
5:04 PM
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Regional Science and Urban Economics 39 (2009) 1–14
Contents lists available at ScienceDirect
Regional Science and Urban Economics
j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / r e g e c
Downtown parking in auto city☆
Richard Arnott a,⁎, John Rowse b
a
b
Department of Economics, University of California, Riverside, 4106, Sproul Hall, Riverside, CA 92521-0427, USA
Department of Economics, University of Calgary, Calgary, AB, Canada T2N 1N4
a r t i c l e
i n f o
Article history:
Received 12 December 2007
Received in revised form 11 July 2008
Accepted 12 August 2008
Available online 20 August 2008
JEL classification:
R40
Keywords:
Parking
Traffic congestion
Parking garages
Parking policy
a b s t r a c t
Arnott and Inci [Arnott, R. and Inci, E., 2006. An integrated model of downtown parking and traffic
congestion. Journal of Urban Economics 60, 418–442] developed an integrated model of curbside parking and
traffic congestion in a downtown area. Curbside parking is exogenously priced below its social opportunity
cost, and the stock of cars cruising for parking, which contributes to traffic congestion, adjusts to clear the
market for curbside parking spaces. Denser downtown areas have garage as well as curbside parking. Because
of economies of scale in garage construction, garages are discretely spaced. The friction of space confers
market power on parking garages. Spatial competition between parking garages, as modeled in Arnott
[Arnott, R., 2006. Spatial competition between downtown parking garages and downtown parking policy.
Transport Policy 13, 458–469], determines the equilibrium garage parking fee and spacing between parking
garages. Also, the stock of cars cruising for parking adjusts to equalize the full prices of curbside and garage
parking. This paper combines the ingredients of these two models, hence presenting an integrated model of
curbside parking, garage parking, and traffic congestion, and examines curbside parking policy in this context
through a numerical example with parameters representative of a medium-sized US city. The central result is
that raising the curbside parking fee appears to be a very attractive policy since it generates efficiency gains
that may be several times as large as the increased revenue raised.
© 2008 Elsevier B.V. All rights reserved.
1. Introduction
Anyone who has parked in the downtown area of a major city
during the business day will attest to its high cost. Parking in a
convenient parking garage is expensive, while finding cheaper
curbside parking normally entails cruising for parking and
walking some distance. To our knowledge, there are no reliable
estimates of the proportion of the average full price of a trip with
a downtown destination that is associated with parking. Informal
estimates of one half seem too high. It seems warranted to say,
however, that economists have paid less attention to downtown
parking than its importance merits. There is a large literature in
economics on urban auto congestion but only a few recent papers
on the economics of downtown parking (which will be reviewed
below).
☆ We would like to thank Eren Inci, Robin Lindsey, David Malueg, and seminar
participants at Clark University, Emory University, the University of California at
Riverside, the University of Colorado at Boulder, the University of Florida at Gainesville,
the University of Massachusetts at Amherst, and the University of California at Irvine for
helpful comments, and Junfu Zhang for pointing out an error in an earlier draft.
⁎ Corresponding author.
E-mail addresses: richard.arnott@ucr.edu (R. Arnott), rowse@ucalgary.ca (J. Rowse).
0166-0462/$ – see front matter © 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.regsciurbeco.2008.08.001
Arnott and Inci (2006) constructed an integrated model of curbside
parking and traffic congestion in an isotropic downtown area with
identical drivers and price-sensitive demand. The curbside meter rate
is set below its social opportunity cost. This results in excess demand
for curbside parking spaces. Parking is saturated, and cars cruise for
parking waiting for a parking spot to open up. The expected time spent
cruising for parking adjusts to clear the market, which is achieved via
adjustment in the density of cars cruising for parking. The cars
cruising for parking contribute to traffic congestion as well. Under
reasonable assumptions, Arnott and Inci demonstrated the existence
and uniqueness of steady-state equilibrium with saturated parking,
and also examined curbside parking policy in the context of the
model.
Denser downtown areas have garage as well as curbside parking.
Because of economies of scale in garage construction, garages are
discretely spaced. The friction of space then confers market power
on parking garages. Arnott (2006) developed a model of spatial
competition between parking garages, which generates an equilibrium parking fee that is above marginal cost. With underpriced
curbside parking and overpriced garage parking, the stock of cars
cruising for parking adjusts to equalize their full prices. This paper
combines the ingredients of these two models (except, to simplify, it
assumes inelastic demand for downtown parking), hence presenting
an integrated model of curbside parking, garage parking, and traffic
2
R. Arnott, J. Rowse / Regional Science and Urban Economics 39 (2009) 1–14
congestion, and examines curbside parking policy in this context
through a numerical example with parameters representative of a
medium-sized, auto-oriented city such as Winnipeg, Perth, San Diego,
Sacramento, or Phoenix.
The addition of garage parking alters the economics of downtown
parking in three interesting ways. First, the equilibrium condition
determining the stock of cars cruising for parking changes. With only
curbside parking, the stock of cars cruising for parking adjusts to clear
the market for trips. In contrast, with both curbside and garage
parking the stock of cars cruising for parking adjusts to equalize the
full prices of curbside and garage parking. Second, even though the
overpricing of garage parking does not create inefficiency directly,
since overall parking demand is assumed to be inelastic, it does so
indirectly in two ways, first, as noted already, by increasing the price
spread between curbside and garage parking and hence the stock of
cars cruising for parking, and second by causing parking garages to be
inefficiently small and too closely spaced. Third, the presence of
garage parking magnifies the distortion associated with the underpricing of curbside parking, or, put alternatively, increases the social
benefit of increasing the curbside parking fee. With only curbside
parking, the equilibrium full price of a downtown trip is determined
by the intersection of the trip demand curve and the curbside parking
capacity constraint. Raising the curbside meter rate does not alter this
full price of trips, but simply converts travel time (which includes intransit and cruising-for-parking time) costs dollar for dollar into meter
revenue, so that every extra dollar of revenue raised increases social
surplus by one dollar. But with garage parking, there is a magnification
effect. Raising the curbside meter rate does not alter the full price of
parking. Raising the curbside meter rate converts cruising-for-parking
time costs dollar for dollar into meter revenue. But there is the added
benefit that the reduction in the stock of cars cruising for parking
reduces traffic congestion, which benefits everyone. In our favored
numerical example, this magnification effect results in a $3.20
increase in social surplus for every dollar increase in meter revenue.
As noted above, the literature on the economics of parking is small.
We start by reviewing the broader literature, and then turn to the
small number of papers that distinguish between curbside and/or
garage parking or analyze cruising for parking.
Early work on the economics of parking argued that parking, like any
other commodity, should be priced at its social opportunity cost (Vickrey,
1954; Roth, 1965). Vickrey (1954) also developed a scheme for demandresponsive pricing of curbside parking. Over the next three decades,
parking was largely ignored by economists, in modal choice studies being
treated simply as a component of the fixed cost of a trip. Modern interest in
the economics of parking started in the early 1990s. Shoup (2005) has led
the way in generating interest in the economics of parking. In the 1990s, he
championed cashing out employer-provided parking, and has considered
many aspects of the economics of parking since then. Arnott et al. (1992)
and Anderson and de Palma (2004) extended the Vickrey bottleneck
model (1969) to analyze the temporospatial equilibrium of curbside
parking when all drivers have a common destination and desired arrival
time, such as for a special event or the morning commute. Arnott and
Rowse (1999) examined the steady-state equilibria of cars cruising for
parking on a circle when parking is unsaturated.
Arnott et al. (2005, Ch. 2, The basic model) presented a model that
examines the interaction between cruising for parking and traffic
congestion with only curbside parking. A more thorough treatment of
that model was provided in Arnott and Inci (2006). Several papers in
the literature have recognized that the stock of cars cruising for
parking adjusts to equalize the full prices of curbside and garage
parking (Calthrop, 2001; Shoup, 2005,1 2006; Arnott et al., 2005;
1
Shoup, Table 11-5, displays the results of 16 studies of cruising for parking in 11
cities over an eighty- year period. The mean share of traffic cruising was 30% and the
average search time was 8.1 min. While the study locations were not chosen randomly,
the results do indicate the potential importance of cruising for parking.
Calthrop and Proost, 2006). Calthrop (2001) and Arnott (2006)
considered the potential importance of garage market power,
Calthrop by assuming a monopoly supplier, Arnott by modeling
spatial competition between parking garages. The Los Angeles model
of Arnott, Rave, and Schöb includes curbside parking, garage parking,
endogenous cruising for parking, and garage market power, but
provides an unpersuasive treatment of garage market power. Arnott
(2006) contained all four elements as well, but focused on the
treatment of garage market power rather than providing a complete
analysis of the model. This paper provides a complete analysis
with the more satisfactory treatment of garage market power, and
also provides calibrated numerical analysis of a variety of parking
policies.
In terms of policy insights, our principal finding – which was
noted above – is that, under conditions of even moderate traffic
congestion, the social benefits from raising curbside parking rates
may be several times the additional meter revenue generated, a
double dividend result. Another important finding is that, with
realistic parameter values, less space should typically be allocated
to curbside parking the larger is the wedge between curbside and
garage parking rates.
Section 2 sets the stage by presenting a simplified model in which
garage parking is provided at constant unit cost. Section 3 presents
and analyzes the central model that takes into account the technology
of garage construction and spatial competition between parking
garages. Section 4 presents calibrated numerical examples for the
central model. Section 5 notes some directions for future research.
And Section 6 provides some concluding comments.
2. A simple model
Understanding the central model of the paper will be facilitated
by starting with a simplified variant. A broad-brush description is
followed by a precise statement.
2.1. Informal model description
The model describes the equilibrium of traffic flow and parking
in the downtown area of a major city.2 To simplify, it is assumed
that the downtown area is spatially homogeneous (isotropic) and
in steady state, and also that drivers are homogeneous. Drivers
enter the downtown area at an exogenous uniform rate per unit
area-time, and have destinations that are uniformly distributed
over it. Each driver travels a fixed distance over the downtown
streets to his destination. Once he reaches his destination, he
decides whether to park curbside or in a parking garage.3 Both
curbside and garage parking are provided continuously over space.
If he parks curbside, he may have to cruise for parking, circling the
block until a space opens up. After he has parked, he visits his
destination for a fixed period of time, and then exits the system.
Garage parking is assumed to be provided competitively by the
private sector at constant cost, with the city parking department
deciding on the curbside meter rate and the proportion of curbside
to allocate to parking. The curbside parking fee (the meter rate) is
less than the garage fee. Consequently, all drivers would like to
park curbside but the demand inflow is sufficiently high that this is
impossible. Curbside parking is saturated (the occupancy rate is
2
The model differs from that in Arnott and Inci (2006) in two respects. Arnott and
Inci consider the situation where all parking is curbside and the demand for trips is
sensitive to the full price of a trip. Here, in contrast, the demand for trips is completely
inelastic, and there is both curbside and garage parking. The model specification is
independent of the form of the street network, but for concreteness one may imagine
that there is a Manhattan network of one-way streets.
3
The paper does not consider parking lots. Parking lots are difficult to treat because
most are transitional land uses between the demolition of one building on a site and
the construction of the next.
R. Arnott, J. Rowse / Regional Science and Urban Economics 39 (2009) 1–14
100%) and the excess demand for curbside parking spaces is
rationed through cruising for parking. In particular, the stock of
cars cruising for parking adjusts such that the full price of curbside
parking, which is the sum of the meter payment and the cost of
time cruising for parking, equals the garage parking payment. The
downtown streets are congested by cars in transit and cruising for
parking. In particular, travel time per unit distance driven increases
with the density of traffic and the proportion of curbside allocated
to parking.
2.2. Formal model
Consider a spatially homogeneous downtown area to which the
demand for travel per unit area-time is constant, D. Each driver travels
a distance δ over the downtown streets to his destination, parks there
for a period of time λ, and then exits.4 A driver has a choice between
parking curbside, where the meter rate is f per unit time, and parking
in a parking garage, at a rate c per unit time, equal to the resource
cost of providing a garage parking space.5 Both curbside and garage
parking are continuously provided over space. By assumption (its
rationale will be given later), f b c, and the excess demand for curbside
parking is rationed through cruising for parking. The stock of curbside
parking is P per unit area, so that the number of garage parking spaces
per unit area needed to accommodate the exogenous demand is
Dλ − P. The technology of traffic congestion is described by the function
t = t(T, C, P), where t is travel time per unit distance, T the stock of cars
in-transit per unit area, and C the stock of cars cruising for parking per
unit area.6 The larger are C and T, the higher the density of traffic on
the city streets, so that tT and tC (subscripts denote partial derivatives)
are positive, and the larger is P, the lower the proportion of street
space available for traffic, so that tP is positive too. It is assumed as well
that t is a convex function of T, C, and P.
D is sufficiently high that, even if all curbside is allocated to parking
(so that P =Pmax), there is still a need for garage parking (i.e., Dλ N Pmax).
Due to the underpricing of curbside parking, the stock of cars cruising
for parking adjusts such that the full price of curbside parking, the sum
of the meter payment and the value of time lost cruising for parking,
equals the full price of garage parking.7 For the moment, it is assumed
that, even when curbside parking is provided free and all curbside is
allocated to parking, the street system can still accommodate the
exogenous demand.8
The density of cars per unit area is T + C, their velocity, v, is 1/t, and
since flow equals density times velocity, the flow in terms of cardistance per unit area-time is (T + C)/t. If there are M distance units of
4
One could equally well assume that each driver travels a distance δ/2 over the
downtown streets to his destination, parks there for a period of time λ, then drives an
equal distance over the streets to his entry point.
5
The paper neglects the subsidization of garage parking, which is practically very
important (Small and Verhoef, 2007, p.113). The bulk of employers, though a smaller
proportion of downtown employers, heavily subsidize their employees' garage
parking. Furthermore, many shoppers receive subsidized garage parking via “parking
validation” whereby a retailer pays for the garage parking of clients who have
purchased goods from his store. The subsidization of garage parking reduces the full
price of garage parking, hence the price differential between curbside and garage
parking, and hence the incentive to cruise for parking.
6
Realistically, travel time per unit distance is affected by cars entering and exiting
curbside parking spaces as well. This could be incorporated into the analysis by adding
the curbside parking turnover rate as an argument to the function t. The specification
of the technology assumes that P is divisible. This is not completely satisfactory since
having only one curbside parking space per block would slow down traffic almost as
much, and perhaps even more, as having an entire curb allocated to parking. This
objection can be accommodated by having a certain proportion of curbside blocks
completely allocated to curbside parking with the remainder being free of curbside
parking. But this would violate the spatial symmetry assumed in the model.
7
The paper ignores possible safety differences between curbside and garage
parking, as well as the search time and walking time inside the parking garage.
8
Primitive conditions for this assumption to hold are given in Section 2.6, which
examines the congestion technology in detail.
3
one-way streets per unit area, then a person standing on a sidewalk
would observe a flow of (T + C)/(Mt) cars per unit time.9 Throughput is
defined analogously to flow but includes only cars in transit10; thus,
the throughput in terms of car-distance per unit area-time is T/t.
Steady-state equilibrium is described by two conditions. The first,
the steady-state equilibrium condition, is that the input rate into the intransit pool, D, equals the output rate, which equals the stock of cars in
the in-transit pool divided by the length of time each car stays in the
pool, T/(δt(T, C, P)):
D¼
T
:
δt ðT; C; P Þ
ð1Þ
This may be written alternatively as Dδ = T/t(T, C, P). Dδ is the input
in terms of car-distance per unit area-time, and T/t(T, C, P) is the
throughput. Let ρ be the value of time.11 The second equilibrium
condition, the parking equilibrium condition, is that the stock of cars
cruising for parking adjusts to equilibrate the full prices of garage and
curbside parking:
cλ ¼ f λ þ
ρCλ
:
P
ð2Þ
The full price of garage parking is cλ. The full price of curbside
parking is f λ plus the (expected) cost of cruising for parking. The
expected time cruising for parking12 equals the stock of cars cruising
for parking, C, divided by the rate at which curbside parking spots
are vacated, P/λ. Thus, holding fixed the expected time cruising for
parking, the stock of cars cruising for parking increases with the
number of curbside parking spaces available. The cost of cruising for
parking equals the expected time cruising for parking times the value
of time. Eq. (2) may be rewritten as
C¼
ðc−f ÞP
;
ρ
ð3Þ
indicating the equilibrium stock of cars cruising for parking as a
function of c, f, P, and ρ.
This simple model has two equations in two unknowns, T and C.
The equations are recursive. Eq. (3) determines C and then Eq. (1)
determines T. Resource costs per unit area-time, RC, are simply ρ(T + C),
the stock of cars in transit and cruising for parking, times the value of
time, plus the resource cost of garage parking, c(Dλ − P).
9
Flow equals density times velocity is known as the Fundamental Identity of Traffic
Flow. Applying this identity in this context requires some care. Ordinarily, density is
measured per unit distance, so that, with velocity measured as distance per unit time,
the dimension of flow is cars per unit time. Here, however, density is measured as cars
per unit area, so that application of the formula gives flow in units of car-distance per
unit area-time. With M miles of city street per unit area, the density in terms of cars
per unit distance is (T + C)/M, and application of the Identity gives flow as cars per unit
time on a street, (T + C)/(Mt).
10
Since the transportation engineering literature has not analyzed situations in
which cars circle the block, it does not make a terminological distinction between flow
and throughput. It seems intuitive to define flow as the number of cars that a
bystander would count passing by per unit time. Throughput too seems an appropriate
choice of term.
11
The large empirical literature on the value of travel time finds it to differ
significantly across travel activities and for in-transit travel to differ according to the
level of congestion. The analysis could be extended straightforwardly to allow for these
considerations.
12
This statement takes into account that the service discipline is random access.
There are several features of the model that can be interpreted as either stochastic or
deterministic. A deterministic interpretation, with FIFO access to curbside parking on
each block, is the most straightforward. Stochastic interpretations are more realistic
but would need to be solved using queuing theory (see, for example, Breuer and Baum,
2005).
4
R. Arnott, J. Rowse / Regional Science and Urban Economics 39 (2009) 1–14
ρ(T + C) + c(Dλ − P). Thus, the constrained social welfare optimization
problem is to choose T, C, and P to
2.3. Full social optimum
The full social optimum entails no cruising for parking. The density
of in-transit traffic is then determined by Eq. (1) with C = 0. The
social welfare optimization problem is to choose T and P to minimize resource costs per unit area-time, subject to the steady-state
equilibrium condition, given by Eq. (1):
min RC ¼ ρT þ cðDλ−P Þ s:t: T−δt ðT; 0; P ÞD ¼ 0:
T;P
ð4Þ
The shadow cost of curbside parking is the increase in in-transit
travel time cost per unit area-time from having one more curbside
parking spot, which, from Eq. (1), is ρdT/dP = ρδtPD/(1 − δtTD)). If, with
P = 0, the shadow cost of curbside parking exceeds c, it is optimal to
allocate no curbside to parking. And if, with P = Pmax, the shadow cost
of curbside parking falls short of c, it is optimal to allocate all curbside
to parking. Otherwise, the optimal P solves
ρδtP D
−c ¼ 0;
1−δtT D
T;C;P
s:t:
iiÞ
Dδt ðT; C; P Þ−T ¼ 0; /
P
ðc−f Þ−C ¼ 0;
’
ρ
ð6Þ
where ϕ is the Lagrange multiplier on constraint i) and φ that on
constraint ii). The second-best optimum may entail no curbside
allocated to parking, in which case there is no cruising for parking,
or all the curbside allocated to parking. An interior optimum is characterized by the first-order conditions:
T : ρ þ /ðDδtT −1Þ ¼ 0
ð7aÞ
C : ρ þ /DδtC −’ ¼ 0
ð7bÞ
P : −c þ /DδtP −
’ðc−f Þ
¼0
ρ
ð7cÞ
Substituting out the Lagrange multipliers yields
ð5Þ
the level of curbside parking should be chosen to equalize the shadow
costs of curbside and garage parking.13
Let ⁎ denote the value of a variable at the social optimum. The
social optimum can be decentralized by setting P = P ⁎, T = T ⁎, and
f = f ⁎ = c.
2.4. Constrained (second-best) social optimum
Since parking policy is local, it is poorly documented and has been
little studied. One empirical regularity stands out, however, at least for
US cities. In downtown areas, the curbside parking fee is considerably
lower than garage parking fees. In downtown Boston, for example, the
meter rate has remained at $1.00 per hour for twenty years, while garage
parking fees are as high as $10.00 for the first hour. What accounts for
this price differential? We have posed this question to several seminar
audiences. The common answer is that downtown merchants' associations lobby City Hall to set the meter rate low in order to draw shoppers
away from suburban shopping centers, where most parking is provided
free. In many downtown shopping areas, merchants pay for shoppers'
garage parking by validating their garage parking stubs. Since most
curbside parking has time limits, and since merchants cannot pay for
their customers' curbside parking, a low meter rate subsidizes the
parking of short-term shoppers. If most curbside parking is occupied by
shoppers, the result is a form of price discrimination. Shoppers end up
paying less for downtown parking than non-shoppers. In this paper, we
do not attempt to model the political economy of downtown parking
policy, but instead simply assume that that the meter rate is set low –
below the marginal cost of a garage parking space – by the local
transportation authority, and explore the implications of the price
wedge between curbside and garage parking.
The constrained social optimum is now considered, where the
constraint is that the curbside parking fee is set below c, with the
stock of cars cruising for parking adjusting so as to satisfy the parking
equilibrium condition. The second-best optimal allocation of curbside to parking minimizes resource costs per unit area, subject to both
the steady-state equilibrium condition, (1), and the parking equilibrium condition, (2). Resource costs per unit area-time are given by
13
iÞ
min RC ¼ ρðT þ C Þ þ cðDλ−P Þ
Recall that the constraint is that the inflow to the in-transit pool equal the outflow.
In conventional traffic flow theory, there are two densities corresponding to a level of
flow. The specification of the minimization problems implies the choice of the lower
density. This complication is addressed in Section 2.5. The convexity of the congestion
function ensures that there is a unique minimum corresponding to the lower density.
ρDδtP
DδtC
þ ðc−f Þ þ
ðc−f Þ−c ¼ 0:
1−DδtT
1−DδtT
ð8Þ
A heuristic derivation is as follows: P should be chosen such that
dRC/dP = 0. From the objective function, dRC/dP = ρdT/dP + ρdC/dP − c;
from constraint ii), dC/dP = (c − f)/ρ; and from constraint i), (dT/dP)
(1 − DδtT) = DδtCdC/dP + DδtP. The last term in Eq. (8) is the marginal
social benefit of P, the reduction in garage costs. The other three
terms are components of the marginal cost, all of which relate to
travel costs. The first term captures the capacity reduction effect; this
is the increase in aggregate in-transit travel costs that comes about
through the reduction in the road space available, holding constant
T and C. The second term captures the cruising-for-parking stock
effect; since the stock of cars cruising for parking is proportional to the
amount of curbside parking, the increase in P increases the stock
of cars cruising for parking, which, holding T and P fixed, increases
congestion. The third term captures the cruising-for-parking congestion effect; via the steady-state equilibrium condition, the increase in
the stock of cars cruising for parking causes the stock of cars in transit
to increase, which further augments congestion.
Let ⁎⁎ denote values at the constrained social optimum. With the
curbside meter rate set at the exogenous level, the constrained social
optimum can be decentralized by setting T = T ⁎⁎, C = C ⁎⁎, and P = P ⁎⁎.
Unless both allocations entail the same corner solution, the optimal
amount of curbside to allocate to parking is greater in the full social
optimum than in the second-best social optimum with underpriced
curbside parking, i.e. P ⁎ N P ⁎⁎. In both allocations, the marginal social
benefit of increasing P by one unit is the saving in garage resource costs,
c. But the marginal social cost of increasing P is lower in the full social
optimum than in the second-best optimum since the costs deriving from
the cruising-for-parking stock and congestion effects are absent.
2.5. Revenue multiplier: the effects of raising the curbside parking fee
A principal theme of the paper is that the underpricing of curbside
parking is wasteful. To formalize this point, this subsection examines
the resource savings from increasing the curbside parking fee by a
small amount when it is below c, holding P fixed. From the expression
for resource costs:
dRC
dT
dC
¼ρ
þρ
df
df
df
dT
dC
¼ρ
þ1
:
dC ð1Þ
df
½ j
ð9Þ
where dT/dC|(1) denotes the change in T associated with a unit
change in C when Eq. (1) is satisfied. Now, the revenue raised from
R. Arnott, J. Rowse / Regional Science and Urban Economics 39 (2009) 1–14
5
the parking fee, R, is Pf, so that dR/df = P. From Eq. (2), ρdC/df = − P.
Thus,
−
½ j
dRC
dT
¼
df
dC
þ1
ð1Þ
dRdf :
ð10Þ
Hence, the resource cost saving per unit area-time from raising the
curbside parking fee equals a multiple of the increase in the parking fee
revenue raised. We term this the multiplication effect. Since the full price
of parking is c, whether a driver parks curbside or in a parking garage,
cruising-for-parking costs fall by exactly the amount of the increase in
parking fee revenue, and there is the added benefit that in-transit travel
costs fall due to the reduction in the stock of cars cruising for parking.
Define
μu−
dRC
dT
df
j
¼1þ
dR
dC ð1Þ
df
ð11Þ
to be the (marginal) revenue multiplier. What determines the size of
the revenue multiplier? Or put alternatively, by how much does a unit
reduction in the stock of cars cruising for parking reduce the total
stock of cars on the road? The answer depends on the technology of
congestion, as well as its level. The revenue multiplier is even larger if
account is taken of the marginal cost of public funds exceeding 1.
2.6. The congestion technology
The steady-state equilibrium condition, (1), can be written
implicitly as C = C(T; P, D). Holding fixed P and D, for each level of T
the function gives the stock of cars cruising for parking consistent with
steady-state equilibrium. Under the assumption that the function t(.)
is convex in T, C, and P, holding P fixed the function C is concave in T.
In the absence of cruising for parking, with realistic congestion
functions there are normally two densities consistent with a given
level of (feasible) flow, i.e. T = δt(T, 0, P)D has two solutions, one
corresponding to regular traffic flow, the other to traffic jam
conditions. Fig. 1 displays the graph of the function C, termed the
steady-state locus, with these two properties. An increase in P causes
the locus to shift down; holding T fixed, Eq. (1) determines an
equilibrium travel time, and to offset the increase in travel time due to
the increase in P requires a decrease in C. An increase in D causes the
locus to shift down; holding P and T fixed, for Eq. (1) to continue to be
satisfied the increase in D must be offset by a decrease in t, and hence
a decrease in C. If D increases sufficiently, there is no (T, C) satisfying
Eq. (1) and a steady-state equilibrium does not exist.
It is assumed furthermore that the function t(.) is weakly separable,
specifically that t = t(T, C, P) = t(V (T, C), P), with V defined as the effective density of cars on the road14 and normalized in terms of in-transit
car-equivalents in the absence of cruising for parking, i.e. V (T, 0) = T.
Define κ(C, P) = maxT T/t(V (T, C), P) to be the throughput capacity of
the street system as a function of C and P, so that κ(0, P) is capacity as
conventionally defined. It indicates the maximum entry rate, in terms
14
In the numerical examples presented later, it will be assumed that congestion
takes the form of a negative linear relationship between velocity and effective density:
v = vf (1 − V/Vj), where vf is free-flow speed and Vj is effective jam density, which is a
slight generalization of Greenshield's Relation. Travel time per unit distance is the
reciprocal of velocity. Then letting t0 denote free-flow travel time, this relationship can
be rewritten as t = t0/(1 − V/Vj). It will also be assumed that a car cruising for parking
creates 1.5 times as much congestion as a car in transit, i.e. V = T + 1.5C. Eq. (1) then
becomes T(Vj − T − 1.5C) = δDt0Vj. Thus, capacity is Vj/(4t0), the effective density
corresponding to this capacity is Vj/2, (dC/dT)(1) = (Vj − 2T − 1.5C)/(1.5T), and the revenue
multiplier is µ = (Vj − 0.5T − 1.5C)/(Vj −2T − 1.5C). We shall assume furthermore that
effective jam density is linearly decreasing in the proportion of curbside allocated to
parking: Vj = Ω(1 − P/Pmax), where Ω is the effective jam density with no curbside
parking.
Fig. 1. Steady-state equilibrium. Notes: 1. C(T;P,D) is the steady-state locus, Eq. (1).
2. C ¼ c−f
ρ P is the parking equilibrium locus, Eq. (2).
of car-distance per unit area-time, the street system can accommodate
in steady-state equilibrium for a given C and P. If Dδ exceeds κ(0, P),
then the entry rate exceeds throughput capacity even in the absence
of cruising for parking, and no steady-state equilibrium exists. If Dδ is
less than κ(0, P), then the steady-state locus lies in the positive
quadrant. The parking equilibrium condition, Eq. (2), can be written as
C = (c − f)P/ρ, giving the equilibrium stock of cars cruising for parking.
Since the condition is independent of T, its graph in Fig. 1, the parking
equilibrium locus, is a horizontal line. If Eq. (2) lies everywhere above
Eq. (1), which occurs if Dδ N κ((c − f) P/ρ, P), the entry rate exceeds
throughput capacity at the equilibrium level of cruising for parking,
and no equilibrium exists.15 If Eqs. (1) and (2) intersect, they do so
twice. A straightforward stability argument16 can be applied to
establish that the left-hand intersection point is stable while the
right-hand one is unstable. Thus, we take the left-hand intersection
point to be the equilibrium.
An initial equilibrium is indicated by E1 in Fig. 1. The revenue
multiplier equals one plus the reciprocal of the slope of the steadystate locus at the equilibrium point. If the curbside parking fee is
lowered, the parking equilibrium locus shifts up, causing the equilibrium to move up along the steady-state locus. Due to the convexity
of the congestion technology, the slope of the steady-state locus at the
equilibrium point falls, and hence the revenue multiplier increases.
Now consider the effect of increasing the amount of curbside allocated
to parking. The steady-state locus shifts down and the parking
equilibrium locus shifts up. The equilibrium T and C increase; effective
density increases, which, along with the decrease in road capacity,
causes traffic congestion to worsen; and the revenue multiplier can be
shown to increase.17
2.7. Complications caused by garage construction technology
The above analysis laid out the economics of equilibrium when
there is curbside and garage parking, when garage parking is priced at
15
Earlier, to avoid considering non-existence of a solution to the resource cost
minimization problems, it was assumed an equilibrium exists even when traffic is as
congested as possible, which occurs when f = 0 and P = Pmax. This condition is that Dδ b
κ(cPmax/ρ, Pmax). With a positive parking fee and/or less curbside allocated to parking, a
solution to the resource cost minimization problems (and the corresponding
equilibria) may exist when this condition is not satisfied.
16
It is reasonable to assume that the parking equilibrium condition is always satisfied,
so that trajectories travel along the Ċ = 0 locus. Ṫ is positive above Ṫ = 0 and negative
below it. If therefore the initial level of T lies below the Ṫ = 0 locus, T decreases, while if it
lies above the Ṫ = 0 locus, T increases. It follows that E1 is a stable equilibrium and E2 an
unstable equilibrium.
^
17
Substituting Eq. (2) into Eq. (1) gives T = δt(T, (c − f)P/ρ, P)D = T (P). Also, from Eq. (3),
^
^
C = (c − f)P/ρ. Then, dT/dC = (dT/dP) ÷ (dC/dP) = [ρ/(c − f)]dT/dP, and so d(dT/dC)/dP = [ρ/(c
^
− f)]d2T/dP2, which can be shown to be positive.
6
R. Arnott, J. Rowse / Regional Science and Urban Economics 39 (2009) 1–14
constant unit cost, and when curbside parking is priced below this
level. Unfortunately, the model is unrealistically simple in assuming
that garage spaces are supplied uniformly over space at constant unit
cost. The technology of garage construction and other factors result in
parking garages being discretely spaced.18 To reduce his walking costs,
a driver is willing to pay a premium to park in the parking garage
closest to his destination. Parking garages therefore have market
power and may exercise it by pricing above marginal cost. Furthermore, spatial competition between parking garages may result in their
being inefficiently spaced. Taking these considerations into account
complicates the economics, since there will then be three distortions
that need to be taken into account, not only the underpricing of
curbside parking but also the overpricing and inefficient spacing of
garage parking.
In the next section, the model of this section is extended to take
into account the exercise of market power caused by the discrete spacing
of parking garages. The exact reason for the discrete spacing of parking
garages is secondary. It is assumed that the discrete spacing arises
from the fixed land area required for a central ramp, which generates
horizontal economies of scale. The optimal spacing minimizes overall
resources costs. The equilibrium spacing is the outcome of spatial
competition between parking garages.
individual curbside parker to cruise for parking on his destination
block, or the spatial homogeneity of traffic flow.21 These assumptions
together imply that the steady-state equilibrium condition for the
simple model, T = δt(T, 0, P)D, continues to hold. Denoting the corresponding equilibrium in-transit density as a function of P by T ⁎(P)
gives TT = ρT⁎ (P).
Efficiency entails identical parking garages being symmetrically
arrayed over space, with diamond-shaped market areas. Let s be the grid
or Manhattan distance between parking garages, x the capacity of each
parking garage, and K(x) the minimum cost per unit time of a garage as a
function of capacity. Each garage services an area of s2/2. With demand
inflow D per unit area-time and parking duration λ, the total number
of parking spaces in a garage's service area is Dλs2/2. Since Ps2/2
curbside parking spaces are provided in the service area, garage capacity
is x = (Dλ −P)s2/2 and GC =K((Dλ −P)s2/2) ÷s2/2. Since the demand for
garage parking is uniformly distributed over space, the average distance
walked by a garage parker is 2s/3 so that average walking time is 2s/(3w),
where w is walking speed, and WC = 2ρs(D −P/λ)/(3w). Combining the
above results gives
RC ¼
2
K ðDλ−P Þ s2
s2
2
þ 2ρs
D− λP
þ ρT⁎ðP Þ:
3w
ð13Þ
3. The central model
The primitives of the model differ from those of the simple model
of the previous section in three respects. First, the garage cost function
incorporates horizontal economies of scale, reflecting the fixed costs
associated with the central ramp. Second, to avoid dealing with price
discrimination based on parking duration, parking duration rather
than visit duration is taken to be exogenous. And third, a grid street
network is assumed.
In many cities, there are both public and private parking garages.
To keep the analysis manageable, however, it is assumed that all
parking garages are private.19 The social optimum is solved first, then
the spatial competition equilibrium is solved when the government
intervenes only through its curbside parking policy.20
3.1. Social optimum
Since travel demand is perfectly inelastic, the social optimum
entails minimizing resource costs per unit area-time. There are three
components to resource costs per unit area-time: garage costs per unit
area-time (GC), walking costs per unit area-time (WC), and in-transit
travel costs per unit area-time (TT):
RC ¼ GC þ WC þ TT:
ð12Þ
It is assumed that the presence of parking garages does not alter
the distance drivers travel over city streets, or the decision of the
18
Suppose, for the sake of argument, that parking garages are continuously
distributed over space. It would be cheapest to construct garage parking on the
ground floor of every building, but this space is especially valuable for retail purposes.
Constructing below-ground parking may be cost effective at the time the building is
constructed, but is expensive for buildings that were originally constructed without
underground parking. Constructing above-ground parking in multi-use buildings
raises structural issues. In most situations, the cost of constructing garage space is
minimized with structures specifically designed as parking garages. Even if parking
were distributed continuously over space, parking garage entrances would not be.