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Modeling Downtown Parking and Traffic Congestion

by Chris Bowman  TP_Bowman

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.

 

 

BackgroundWe 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

TB_1

Adding the cost of walking from location, x, to the CBD,

TB_2

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):

TB_3

IfTB_Xis the furthest distance parked, the car parked at this location has the smallest search cost, γ, so that:

TB_4

By equating the supply and demand for parking, which requires

TB_5

We can solve for the equilibrium expected cost, c, in implicit form:

TB_6

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.

TB_7

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

TB_8

Therefore, the optimal number of people parking at x is presented as

TB_9

The population constraint can then be shown as

TB_10

And after integrating the left hand side,

TB_11

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

TB_12

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

TB_13

By inserting the optimal parking density no(x) expression (9) into the above equation, we reach the optimal parking fee

TB_14

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.


*
Shoup, Donald C. “Cruising for parking.” Transport Policy 13.6 (2006): 479-486.


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