Chris Bowman
Technical Review
Modeling Downtown Parking and Traffic Congestion
A Model by: Anderson, Simon P., and Andre De Palma. "The economics of pricing
parking." Journal of Urban Economics 55.1 (2004): 1-20.
Background: We have all experienced the frustration of trying to find a parking spot in a
crowded city. When on-street parking is free or the same price throughout the city, we
try to park closest to our destination. However, the congestion resulting from everybody
trying to park closest to the CBD creates a parking pattern that is less than socially
optimal.
In this model, we are dealing only with individuals that do not have assigned parking
spaces in the city, such as shoppers. We will initially make the following assumptions:
I.
All individuals are traveling to a common location at x=0 (the CBD).
II.
The CBD is located at the end of a long, narrow city, and is served by parallel
access roads. Perpendicular to the access roads are side streets that are used for
on-street parking. Cars can park on street at any free location.
III.
There are N individuals located far away.
IV.
Each individual first drives at speed vd into downtown, then begins looking for an
empty parking spot on a side street.
V. Once an available parking spot is found, the individual walks at speed vw to the
CBD.
VI.
The more people trying to park, the longer it takes to find a spot.
Once in the city, the individual will stop at some distance, x, from the CBD and search
for an open spot while incurring a cost γ per lot inspected. He will then walk to the CBD.
The total number of parking spots on the interval [x,x + dx] is represented by K(x) dx ,
with K(x) = k . Therefore, the city has width k, with the CBD located at the end. The
number of occupied parking spots over the interval [x,x + dx] is denoted by n(x) dx
with n(x)  k. The probability that a randomly tested spot will be free is given by
q(x) = [k −n(x)]/k .
Basic Model
The expected cost of an individual searching for an open spot at location x is
Adding the cost of walking from location, x, to the CBD,
Where t is the net dollar cost of walking as opposed to driving.
In an equilibrium where parking is unpriced, all parking locations have the same expected
cost, c. Rearranging (2):
If is the furthest distance parked, the car parked at this location has the smallest search
cost, γ, so that:
By equating the supply and demand for parking, which requires
We can solve for the equilibrium expected cost, c, in implicit form:
Introduction of a Social Planner
If we introduce a social planner who wants to provide the socially optimal parking
pattern, she will want to minimize the social cost of getting the individuals to the CBD.
The solution {n(x), xo} to the optimal control problem above involves equating marginal
social cost (with respect to n(x) ) for all locations where at least one car is parked. By
differentiating the integrand above, we solve for the marginal social cost λ
Therefore, the optimal number of people parking at x is presented as
The population constraint can then be shown as
And after integrating the left hand side,
Rewriting this last equation using the value of λ in (8), k/t (√λ −√γ )2 = N .
The optimized marginal social cost is therefore λ = (√γ +√Nt/k )2 . This tells us the
optimal value of the location of the last parking place, which we can compare to the
location of the last parking spot under the model without a social planner. Substituting
this value of λ in Eq. (8) leads to
Discussion
Comparing the findings from the unpriced parking scenario in Eq. (4) and (6) with the
optimal results achieved by the social planner (10), we can see that under the optimal
scenario, the parking span is larger. This means that when there is unpriced parking,
parking becomes more tightly spaced than is socially optimal. This results from the fact
that free parking is a common property resource, and that people do not take into account
that, by deciding to search for a spot closest to the CBD, they are increasing the search
costs of others. While this may seem intuitive, it sets up a basis for studying urban
parking fee structures. By raising the cost of on-street parking closest to the CBD, a city
may be able to reduce some of the congestion associated with parking. This is added to
the model by stating that the optimal parking fee, τ(x), occurs when the parking price is
equal to the difference between marginal social and private cost, given as
By inserting the optimal parking density no(x) expression (9) into the above equation, we
reach the optimal parking fee
This equation for the optimal parking fee demonstrates that the fee should decrease as
distance from the CBD increases.
Extensions
Because free or wrongly priced unassigned urban parking spots lead to increased
congestion and tighter parking closer to the CBD, there are a number of possible
investigations or solutions to the problem that ought to be considered. This model does
not take into account the time limits placed on many metered parking spots, or account
for the length of stay once parked. This factor may be important in determining optimum
pricing strategies across the urban landscape. Also, the model should consider an
individual’s option of choosing not to search for a spot, but instead to pay a premium to
park immediately in a garage. Shoup (2006)* explores the relationship between the point
of indifference between cruising for a spot, and paying to park in a garage.
Because there are so many factors involved in individuals’ parking decisions, it is
difficult to build an entirely comprehensive model, but this model provides a base to
begin understanding parking price theory.
With the dawn of smart phones, it may soon be possible to virtually assign spots to
individuals driving into urban areas, and a bidding scheme could be established to set the
price. It would be a difficult system to enforce, but an interesting thing to consider.
*
*
Shoup, Donald C. "Cruising for parking." Transport Policy 13.6 (2006): 479-486.