ecn741: urban economics the urban transportation system

ECN741: Urban EconomicsThe Urban Transportation System

The Urban Transportation System

Class Outline

Models with street grids

Models with street grids and arteries

Other models

Introduction to traffic congestion


Street Grids

The assumption that streets are rays coming out from the center can easily be replaced by the assumption that there is a street grid.

This point was clearly made in Alonso, who presents the intuition and examples of iso-cost lines but does not introduce a grid into a formal urban model.

The trick is to use the assumption of an infinitely dense grid to make the math work and to approximate a discrete set of vertical and horizontal streets.


Alonso Version of Simple Grid


Modeling a Simple Grid

In this set-up people travel to work along horizontal and vertical streets.

The continuous math literally means that the distance from their house to the nearest street is ignored.

In the simplest case, transportation costs equal cost per mile multiplied by “Manhattan distance.” t x y


Modeling a Simple Grid, 2

In the positive quadrant, this set-up implies that

An iso-cost line is the set of points with equal transportation cost.

The above figures imply that y = u, so T = tu and an iso-cost line is defined by

or T

T t x y y xt

y u x



This equation defines a square iso-cost line; the x-intercept is also at u.

Thus the iso-cost line in the positive quadrant is the hypotenuse of right triangle and its length, which is ¼ of land supply or L{u}, is

2 2 22 2

4 4 2

d u u u u

L u d u



Recall that when we solved a basic urban model, we assumed that L{u} = 2πu.

So all we have done is replace 2π with 4√2.

The only thing we have changed is the “land constant.”

All the equations and comparative static results still hold!



But we have substantially altered the geography and characteristics of the urban area.

Because population is proportional to the land constant, we can compare the populations in a radial city and a grid city.

A radial city has 11.1% more people.

21.111

4 2RADIAL

GRID

N u

N u



We can also compare the physical size of the two cities by comparing the area inside the iso-cost lines that go through .

A radial city has 57.1% greater area than a grid city.

u

2

2

Area1.571

Area 2 2RADIAL

GRID

u

u



Putting these two results together, we can also compare population densities in the two cities.

Density equals population divided by area, so .

A radial city has 41.4 percent more people per square mile than does a grid city.

2

2

/ 2 / 2 1.5711.414

1.1112 / 4 2 2 / 4 2RADIAL

GRID

D u u

D u u


Implications for Empirical Work

d1 d2


Implications for Empirical Work

Note that

Thus, if the street network is actually like a grid and the empirical work uses radial distance to the CBD, distance measures are too high by as much as 41%!

2 1

1 1and 2

2 2d u d d u

2

1

1 22 1.414

1 222

d

d


Varying Transportation Costs

The next step is to allow transportation costs to vary on the vertical and horizontal streets.

So in the positive quadrant

h vT t x t y

; slopeh h

v v

T t x ty

t t


Varying Transportation Costs, 2

Note that in this figure,

th < tv

and the city shape stretches in the low-cost direction.

d



Now anchoring an iso-cost on the vertical axis, we have

The iso-cost line runs from (0,u) to ((tv/th)u, 0).

v h v

h

v

T t u t x t y

ty u x

t



So the length of an is-cost line is:

And the land constant is2

4 4 1 v

h

td

t

2 2

2 1v v

h h

t td u u u

t t



Recall that the population of a radial city is 11.1% larger than the population of a grid city with the same t.

How much lower would th have to be for a “stretched” grid city to have the same population as the radial city?

The two land constants, and hence the two populations are the same if

2

2

2 4 1

1 1.2112

h

h

t

t

t

t



It follows that th would have to equal t divided by 1.211,

which is the same as saying that th would have to 17.42% smaller than t. ( 1-(1/1.211) = 1- 0.8258 = .1742)

In other words, a grid city’s “disadvantage” in using space caused by its longer commuting distances can be offset by lowering travel costs in one direction by 17.42%.

As an exercise: Show that if N is equal in the two cities, the radial city’s land area is greater by 29.8%.


Grids Plus Arteries

The next step in the analysis is to add commuting arteries, which could be freeways, subways, or trains.

The analysis depends on the relationship between the arteries and the grid.

The two simplest cases are in the following graphs.


Grids Plus Arteries, 2


Alonso Version of Grid with Arteries


Commuting Sheds

The introduction of grids adds a key new concept, namely a commuting “shed.”

A commuting shed is analogous to a water shed;

People on one side of a boundary commute in one direction, whereas people on the other side commute the other way

Just like water that falls on one side of the Continental Divide flows to the Pacific Ocean, whereas water that falls on the other side flows to the Atlantic Ocean.


Commuting Sheds, 2

Shed boundaries

Commuting paths

Artery


Vertical and Horizontal Arteries

This shed picture describes vertical and horizontal arteries.

The shed boundary is defined by

In the positive quadrant, this is

a h a v

a h

a v

t y t x t x t y

t ty x

t t


Vertical and Horizontal Arteries, 2

The iso-cost is defined by

or

a hT t y t x

h

a a

tTy x

t t



As before, define T = tau; the iso-cost becomes

The slope is less than (-1) because th > ta.

a h h

a a

t u t x ty u x

t t


Vertical and Horizontal Arteries, 4(0,u)

dPoint that is on shed and iso-cost line=(x*,y*)City stretches out along arteries.



Pooling earlier results for shed and iso-cost:

Solving for (x*,y*)

a h h

a v a

t t ty x u x

t t t

2 2

( ) ( )* and *

( ) ( )a a v a a h

a h v a h v

t t t t t tx u y u

t t t t t t



Finally, plugging both points into the standard distance formula, we find that

2 2

2 22 2

2 2 2

2 2

( ) ( )1

( ) ( )

(( ) ( ) )( )

(( ) )

a a v a a h

a h v a h v

a v a v

a h v

t t t t t td u u

t t t t t t

t t t tu

t t t



Once again, we find that extending the transportation system does not change the algebra of the basic model, although it does change the map.

If tv = th, the new land constant is simply 8d, and all the equations and comparative statics from the basic model still hold.

If tv ≠ th , then the formula for the length of the iso-cost line below the artery switches the tv and th terms and the length is 4 times the sum of the two lengths.


Diagonal Arteries

The next case to consider is diagonal arteries.

We will keep it simple, with the same costs on vertical and horizontal streets = tg

And with arteries at a 45 degree angle to the grid.


Diagonal Arteries, 2(0,u)

d

Artery

Shed Boundary

y - x

x√2


Diagonal Arteries, 3

Thus, the equation for an iso-cost line is

or

( ) 2g g aT t u t y x t x

1 2a

g

ty u x

t



Note that the slope of the iso-cost line in the positive quadrant could be positive as drawn, yielding a star shape, or negative, yielding a hexagon.

The hexagon case arises if:

1 2 0 or 2ag a

g

tt t

t



Because y = x on the artery, we can write

This gives us the coordinates of the point on the artery, and we can solve for the distance.

2g

a

tx y u

t

2

{ } 8 1 2g g

a a

t tL u u

t t



Diagonal arteries do not have to be at a 45 degree angle, of course.

My article shows how to find the land constant with any number of arteries set at any angles to the grid.

In these cases, one can have travel in the “wrong” direction because the savings from getting to the artery faster offset the “backwards” direction.

The analysis builds on the following graphs:


Arteries and Maps

As indicated earlier, adding arteries changes the map of an urban area: the area extends farther from the center along a high speed artery (or, for that matter, along high-speed streets).

This impact can be observed in many places, as development extends along main streets or highways.

One well-known urban scholar, Nate Baum-Snow, uses this point to argue that the building of arteries caused suburbanization. (N. Baum-Snow, “Did Highways Cause Suburbanization?”, QJE 2007)


Arteries and Maps, 2

The Baum-Snow work is interesting and well done.

But in a theory paper on which his QJE paper draws (“Suburbanization and Transportation in the Monocentric Model,” JUE, 2007), he says:

The main innovation of this model is that it incorporates highways into the transportation infrastructure of a monocentric city. Highways are modeled as linear “rays” emanating from the city’s core along which the travel speed is faster than on surface streets.

He does not cite my paper, even though it appeared in the same journal 14 years before—and does exactly the same thing!.

Somebody needs to tell him about Google scholar!


Other Forms of Arteries

These methods can be applied to many other types of arteries.

This point was clearly recognized by Alonso (whose contributions are also not mentioned by Baum-Snow).

The next two slides give one alternative arrangement in my article and another from Alonso’s book.


City with Several Vertical Arteries


Alonso: Arteries and Square Beltway


Anas-Moses

A final type of transportation system to discuss is one devised by Anas and Moses (JUE, 1979).

They assume arteries and city streets.

They assume that travel on city streets can be approximated by travel on a circle.

They show that people making different mode choices will sort into different locations.


The Anas-Moses Transportation System

Artery

Circular Street House

ρ

u

Shed Boundary


Congestion

Congestion is a very difficult topic.

Nobody has solved an urban model with a general treatment of congestion.

The problem is a fundamental simultaneity:

The population at a location depends on t,

But t depends on the population that commutes through a location.


Congestion, 2

One approach is to assume that the person at the outer edge of the city leaves home first, and all people inside that location join her as she passes by.

So all people on a given ray commute together (and arrive at work at exactly the same time).

Congestion at a give u therefore depends on the number of households living outside u:

{ } { }u

u

t u f N u du


The Standard Model of Congestion

Radial Street

Outer Edge of City Transportatio

n costs at u depend on

u

{ }u

u

N u du


Congestion, 3

Solow (Swedish J. of Econ., 1972) manages to solve a model like this assuming a Cobb-Douglas utility function with a housing exponent of ½.

This approach can be found in many simulation models, including Mills’ 1972 book.

But this approach is contradictory, because any commuter can save time by waiting until others have passed and then catching up with them.


Congestion, 4

Another approach is to have a street grid with a single vertical artery and assume that people commute in cohorts (Yinger, JUE, 1993).

This model can be solved, but also has the “wait until later” problem.

The iso-cost lines and bid functions are illustrated in the next 2 slides; c/m is the extent to which having more commuters lowers commuting speed divided by road width.

ecn741: urban economics the urban transportation system

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