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Urban growth boundaries: An effective second-best remedy for unpriced traffic congestion?

By Katerina Valtcheva  TP_Katerina


For this technical presentation I have chosen Jan Brueckner’s paper from 2007 titled “Urban growth boundaries: An effective second-best remedy for un-priced traffic congestion?” In this paper he explores the welfare gain from using UGB as a tool for reducing the distortions created by a first-best toll regime in a congested city and, specifically, un-priced traffic congestion. His paper is based on previous scholarly works, which have proven that appropriate urban growth boundaries (UGB) improve welfare: but what Brueckner is interested in his paper is quantifying this welfare gain. Brueckner’s findings show that UGB are not an effective substitute for a first-best toll regime, nor are they effective as a second-best method in a congested city. These findings are relevant for cities that satisfy the assumptions of the standard monocentric city model.

Brueckner uses Anas and Rhee’s 2006 paper “Curbing excess sprawl with congestion tolls and urban boundaries” as a starting point for his work. In their paper, the two economists numerically evaluate congestion tolls and UGB in a congested city. Their findings show something rather different than previous works on the subject. It turns out that when a city substantially differs from the standard monocentric model used by Brueckner, a toll regime is shown to increases welfare, while the use of UGB, in fact, reduces welfare. What make their result so divergent from previous findings are their assumptions that the city has dispersed employment, where trips within the city are explained by commuting as well as shopping, and “consumer location choices are influenced by random idiosyncratic preferences” (Brueckner 2007).

In his paper, Brueckner makes assumptions based on the standard framework used for a congested monocentric city. Some important assumptions utilized include Cobb-Douglass preferences of consumers over housing consumption and the radial symmetry of the city with a constant fraction of the land used for housing and the rest for radial roads. In addition, he follows a paper from 1985 by Pines and Sadka when he assumes that the city is fully closed with each resident receiving an equal fraction of the aggregate residential rent revenue, which allows him “to conduct a straightforward welfare analysis” (Brueckner 2007). He also assumes that the revenue from tolls is equally redistributed to consumers.

The author defines a number of equations, which express the number of residents that live beyond a certain distance from the central business district (CBD) (Fig. 1), the cost per mile of commuting at a certain distance (Fig. 2), the congestion toll per mile at this distance (Fig. 3), and the commuting cost from that distance (Fig. 4), respectively.


Figure 1


Figure 2


Figure 3


Figure 4

By differentiating the equations in figure 1 and figure 4, Brueckner obtains two relationships (Fig. 5).


Figure 5

The first one shows how, as the distance from CBD increases, the population that lives at a greater distance decreases at the same rate; the second relationship shows that as this same distance increases, so do commuting cost at a rate that equals the cost of direct commuting at that distance along with the cost of the toll. Finally, Brueckner shows that income must equal the sum of exogenous income, income from rent and the income from the toll. It is important to note that in Brueckner’s analysis the “density in any one location in the city depends on densities at all other locations” (Brueckner 2007) as in his model his city is congested. As an implication, instead of solving a set of static simultaneous equations, Brueckner uses an iterative procedure that relies on the equations shown on figure 5 to find the equilibrium of the system.

The results from the method previously described allow Bruecker to compare the welfare generated in a laissez-faire equilibrium, a toll-regime equilibrium, and an equilibrium under an optimal urban growth boundary. As he compares the equilibria generated under four cases of different parameters, he finds that UGBs fail to increase central density in congested cities. This failure to noticeably raise densities around centers of employment is the main reason Brueckner gives for the poor performance of the method. As this is unlikely to change even if a city has more than one such center, Brueckner believes that his findings will hold even if he did not use the monocentric city model in his study. This is one area where I see potential for further improvement of the paper and a strengthening of its implications via performance of numerical analysis for a city with multiple employment centers as well as performance of the same analysis when adding other features to the model. Proving that UGBs are not an effective method of reducing distortions caused by a first-toll regime regardless of the assumptions used to model a city would increase the scope of Brueckner’s findings.

Brueckner’s paper is important as it quantifies the results from previous papers and, thus, takes the knowledge about UGB a step further. Without it a city could have wasted time and resources setting a growth boundary in hope of reducing un-priced traffic congestion only to find out that despite the fact that UGBs generally increase welfare, they are ineffective when used for the previously mentioned purpose.


Brueckner, Jan K. “Urban growth boundaries: An effective second-best remedy for unpriced traffic congestion?.” Journal of Housing Economics 16.3 (2007): 263-273.

Anas, A., Rhee, H.-J., 2006. Curbing excess sprawl with congestion tolls and urban boundaries. Regional Science and Urban Economics 36, 510–541.


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


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):


IfTB_Xis 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



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.



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.