Urban Transportation Systems Through Modeling
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ECN741: Urban Economics
The Urban
Transportation System
Professor John Yinger, The Maxwell School, Syracuse University, 2018
The Urban Transportation System
Class Outline
1. Models with street grids
2. Models with street grids and arteries
3. Other models
4. Introduction to traffic congestion
The Urban Transportation System
Class Outline
1. Models with street grids
2. Models with street grids and arteries
3. Other models
4. Introduction to traffic congestion
The Urban Transportation System
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 made in Alonso, who presents the
intuition and examples of iso-cost lines with a grid but
does not introduce a grid into a formal urban model.
The trick (Yinger,
JUE
, May 1993) 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.
The Urban Transportation System
The Urban Transportation System
Alonso Version of Simple Grid
The Urban Transportation System
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.”
The Urban Transportation System
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
The Urban Transportation System
Modeling a Simple Grid, 4
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
The Urban Transportation System
Modeling a Simple Grid, 5
Recall that when we solved a basic urban model, we
assumed that
L
{u
} = 2
π
u
.
So all we have done is replace
2
π
with
4√2.
The only thing we have changed is the “land
constant.”
All the equations and comparative static results still
hold!
The Urban Transportation System
Modeling a Simple Grid, 6
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.
The Urban Transportation System
Modeling a Simple Grid, 7
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.
The Urban Transportation System
Modeling a Simple Grid, 8
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.
The Urban Transportation System
Implications for Empirical Work
d
2
d
1
d
3
The Urban Transportation System
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 small by as much as 29.3%!
The Urban Transportation System
Varying Transportation Costs
The next step is to allow transportation costs to vary on
the vertical and horizontal streets.
So in the positive quadrant
The Urban Transportation System
Varying Transportation Costs, 2
Note that in
this figure,
t
h
<
t
v
and the city
shape stretches
in the low-cost
direction.
d
The Urban Transportation System
Varying Transportation Costs, 3
Now anchoring an iso-cost on the vertical axis, we have
The iso-cost line runs from
(0,
u
)
to
((
t
v
/t
h
)
u
, 0)
.
The Urban Transportation System
Varying Transportation Costs, 4
So the length of an is-cost line is:
And the land constant is
The Urban Transportation System
Varying Transportation Costs, 5
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
t
h
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
The Urban Transportation System
Varying Transportation Costs, 6
It follows that
t
h
would have to equal
t
divided by 1.211,
which is the same as saying that
t
h
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%.
The Urban Transportation System
Class Outline
1. Models with street grids
2. Models with street grids and arteries
3. Other models
4. Introduction to traffic congestion
The Urban Transportation System
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.
The Urban Transportation System
Grids Plus Arteries, 2
The Urban Transportation System
Alonso Version of Grid with Arteries
The Urban Transportation System
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.
The Urban Transportation System
Commuting Sheds, 2
Shed boundaries
Commuting paths
Artery
The Urban Transportation System
Vertical and Horizontal Arteries
This shed picture describes vertical and horizontal
arteries.
The shed boundary is defined by
In the positive quadrant, this is
The Urban Transportation System
Vertical and Horizontal Arteries, 2
The iso-cost is defined by
or
The Urban Transportation System
Vertical and Horizontal Arteries, 3
As before, define
T = t
a
u
; the iso-cost becomes
The slope is less than (-1) because
t
h
> t
a
.
The Urban Transportation System
Vertical and Horizontal Arteries, 4
(0,
u
)
d
Point that is on shed
and iso-cost line
=(
x*,y*
)
City stretches out
along arteries.
The Urban Transportation System
Vertical and Horizontal Arteries, 5
Pooling earlier results for shed and iso-cost:
Solving for (
x*,y*
)
The Urban Transportation System
Vertical and Horizontal Arteries, 6
Finally, plugging both points into the standard distance
formula, we find that
The Urban Transportation System
Vertical and Horizontal Arteries, 7
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
t
v
= t
h
, the new land constant is simply
8
d
, and all the
equations and comparative statics from the basic
model still hold.
If
t
v
≠
t
h
, then the formula for the length of the iso-cost
line below the artery switches the
t
v
and
t
h
terms and
the length is 4 times the sum of the two lengths.
The Urban Transportation System
Diagonal Arteries
The next case to consider is diagonal arteries.
We will keep it simple, with the same costs on vertical
and horizontal streets =
t
g
And with arteries at a 45 degree angle to the grid.
The Urban Transportation System
Diagonal Arteries, 2
(0,
u
)
d
Artery
Shed Boundary
y - x
x
√2
The Urban Transportation System
Diagonal Arteries, 3
Thus, the equation for an iso-cost line is
or
The Urban Transportation System
Diagonal Arteries, 4
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:
The Urban Transportation System
Diagonal Arteries, 5
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.
The Urban Transportation System
Diagonal Arteries, 6
Diagonal arteries do not have to be at a 45 degree
angle, of course.
Yinger (
JUE
, May 1993) also 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:
The Urban Transportation System
The Urban Transportation System
The Urban Transportation System
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
, May 2007)
The Urban Transportation System
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
, November 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 Yinger (
JUE
, May 1993), even though it appeared
in the same journal 14 years before—and does exactly the same
thing!.
Make sure you use Google scholar!
The Urban Transportation System
Class Outline
1. Models with street grids
2. Models with street grids and arteries
3. Other models
4. Introduction to traffic congestion
The Urban Transportation System
Other Forms of High-Speed Modes
These methods can be applied to many other types of
high-speed modes that do not go to the CBD.
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
Yinger and another from Alonso’s book.
The Urban Transportation System
City with Several Vertical Arteries
The Urban Transportation System
Alonso: Arteries and Square Beltway
The Urban Transportation System
Anas-Moses
A final type of transportation system to discuss is one
devised by Anas and Moses (
JUE
, April 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 (or,
equivalently, because all people are alike, people who
live in different places will make different mode
choices).
The Urban Transportation System
The Anas-Moses Transportation System
Artery
Circular Street
House
ρ
u
Shed Boundary
The Urban Transportation System
Class Outline
1. Models with street grids
2. Models with street grids and arteries
3. Other models
4. Introduction to traffic congestion
The Urban Transportation System
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.
The Urban Transportation System
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
:
The Urban Transportation System
The Standard Model of Congestion
Radial Street
Outer Edge of City
Transportation
costs at u
depend on
u
The Urban Transportation System
Congestion, 3
Solow (
Swedish J. of Econ.
, March 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.
The Urban Transportation System
Congestion, 4
Another approach is to have a street grid with a single
vertical artery and assume that people commute in a
cohort, defined as a set of people who take the same
time to get to work (Yinger,
JUE
, September 1993).
This model can be solved, but also has the “wait until
later” problem—at least for the last cohort.
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.
The Urban Transportation System
The Urban Transportation System
The Urban Transportation System
Congestion, 5
One possible solution to the “wait until later” problem,
which has not yet appeared in an article, is to assume that
every commuter expects other commuters to make the same
decisions she does.
If a commuter realizes she can wait until other traffic has left,
she will assume that other commuters will wait, too.
Hence, commuters at the outer edge wait until they know they
can just get to work commuting with their entire cohort.
This outcome appears to be an equilibrium, because any
commuter in the outer cohort who waits a little longer to leave
for work will quickly be joined by other commuters—and hence
will be late.
A formal game-theoretic analysis of this case awaits!
Explore the dynamics of urban transportation systems through various models, from simple grids to more complex structures. Learn how transportation costs are calculated, and how different grid layouts impact travel efficiency. Dive into the theory behind traffic congestion and the implications for urban planning and development.
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Presentation Transcript
ECN741: Urban Economics The Urban Transportation System Professor John Yinger, The Maxwell School, Syracuse University, 2018
The Urban Transportation System Class Outline 1. Models with street grids 2. Models with street grids and arteries 3. Other models 4. Introduction to traffic congestion
The Urban Transportation System Class Outline 1. Models with street grids 2. Models with street grids and arteries 3. Other models 4. Introduction to traffic congestion
The Urban Transportation System 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 made in Alonso, who presents the intuition and examples of iso-cost lines with a grid but does not introduce a grid into a formal urban model. The trick (Yinger, JUE, May 1993) 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.
The Urban Transportation System Alonso Version of Simple Grid
The Urban Transportation System 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
The Urban Transportation System Modeling a Simple Grid, 2 In the positive quadrant, this set-up implies that ( T t x y = + T t ) or = y x 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 y u = x
The Urban Transportation System Modeling a Simple Grid, 4 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 of land supply or L{u}, is = + = = 2 2 2 2 2 d u u u u ( ) L u = = 4 4 2 d u
The Urban Transportation System Modeling a Simple Grid, 5 Recall that when we solved a basic urban model, we assumed that L{u} = 2 u. So all we have done is replace 2 with 4 2. The only thing we have changed is the land constant. All the equations and comparative static results still hold!
The Urban Transportation System Modeling a Simple Grid, 6 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. N N 2 u = = 1.111 RADIAL 4 2 u GRID A radial city has 11.1% more people.
The Urban Transportation System Modeling a Simple Grid, 7 We can also compare the physical size of the two cities by comparing the area inside the iso-cost lines that go through . u = 2 Area Area u u = = 1.571 RADIAL 2 2 2 GRID A radial city has 57.1% greater area than a grid city.
The Urban Transportation System Modeling a Simple Grid, 8 Putting these two results together, we can also compare population densities in the two cities. Density equals population divided by area, so . / 2 2 / 4 2 GRID D u 2 / 2 1.571 1.111 RADIAL D u u = = = = 1.414 2 2/ 4 2 u A radial city has 41.4 percent more people per square mile than does a grid city.
The Urban Transportation System Implications for Empirical Work d2 d3 d1
The Urban Transportation System Implications for Empirical Work Note that = = + = 2 2 2 2 , , and ( ) (2 ) ( ) d d u d d d d d u 1 2 3 2 3 2 1 Thus, if the street network is actually like a grid and the empirical work uses radial distance to the CBD, distance measures are too small by as much as 29.3%! measured true 1 d d u = = = = 0.707 1 2 2 2 u 2
The Urban Transportation System Varying Transportation Costs The next step is to allow transportation costs to vary on the vertical and horizontal streets. T t x = + t y h v So in the positive quadrant T t x t = = ;slope h h y t t v v
The Urban Transportation System Varying Transportation Costs, 2 Note that in this figure, d th < tv and the city shape stretches in the low-cost direction.
The Urban Transportation System Varying Transportation Costs, 3 Now anchoring an iso-cost on the vertical axis, we have T t u = = t x t y + v h t t v = h y u x v The iso-cost line runs from (0,u) to ((tv/th)u, 0).
The Urban Transportation System Varying Transportation Costs, 4 So the length of an is-cost line is: 2 2 t t t t = + = + 2 1 v v d u u u h h And the land constant is 2 t t = + 4 4 1 v d h
The Urban Transportation System Varying Transportation Costs, 5 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 thhave 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 t = + 2 4 1 t h 2 t = = 1 1.211 2 t h
The Urban Transportation System Varying Transportation Costs, 6 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%.
The Urban Transportation System Class Outline 1. Models with street grids 2. Models with street grids and arteries 3. Other models 4. Introduction to traffic congestion
The Urban Transportation System 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.
The Urban Transportation System Grids Plus Arteries, 2
The Urban Transportation System Alonso Version of Grid with Arteries
The Urban Transportation System 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.
The Urban Transportation System Commuting Sheds, 2 Shed boundaries Commuting paths Artery
The Urban Transportation System Vertical and Horizontal Arteries This shed picture describes vertical and horizontal arteries. The shed boundary is defined by t y + = + t x t x t y a h a v In the positive quadrant, this is t t t t = a h y x a v
The Urban Transportation System Vertical and Horizontal Arteries, 2 The iso-cost is defined by = T t y t x a h or t t T t = h y x a a
The Urban Transportation System Vertical and Horizontal Arteries, 3 As before, define T = tau; the iso-cost becomes t u y t t x t t = = a h h u x a a The slope is less than (-1) because th > ta.
The Urban Transportation System Vertical and Horizontal Arteries, 4 (0,u) d Point that is on shed and iso-cost line =(x*,y*) City stretches out along arteries.
The Urban Transportation System Vertical and Horizontal Arteries, 5 Pooling earlier results for shed and iso-cost: t y x t t t t t = = a h h u x a v a Solving for (x*,y*) ( ) ( ) t t t t t t = = * and * a a v a a h x u y u 2 2 ( ) a t ( ) a t h v t t h v t t
The Urban Transportation System Vertical and Horizontal Arteries, 6 Finally, plugging both points into the standard distance formula, we find that 2 2 ( ) ( ) t t t t t t = + 2 2 1 a a v a a h d u u 2 2 ( ) a t ( ) a t h v t t h v t t + 2 2 2 (( ) t ( ) )( v t ) t t = a a v u 2 2 (( ) t ) h v t t a
The Urban Transportation System Vertical and Horizontal Arteries, 7 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 tvand th terms and the length is 4 times the sum of the two lengths.
The Urban Transportation System 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.
The Urban Transportation System Diagonal Arteries, 2 (0,u) Artery d y - x x 2 Shed Boundary
The Urban Transportation System Diagonal Arteries, 3 Thus, the equation for an iso-cost line is = = + ( ) 2 T t u t y x t x g g a or t t = + 1 2 a y u x g
The Urban Transportation System Diagonal Arteries, 4 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: t t 1 2 0 or 2 a t t g a g
The Urban Transportation System Diagonal Arteries, 5 Because y = x on the artery, we can write t g = = x y u 2 t a This gives us the coordinates of the point on the artery, and we can solve for the distance. 2 t t t t g g = + { } 8 L u 1 2 u a a
The Urban Transportation System Diagonal Arteries, 6 Diagonal arteries do not have to be at a 45 degree angle, of course. Yinger (JUE, May 1993) also 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:
The Urban Transportation System 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, May 2007)
The Urban Transportation System 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, November 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 Yinger (JUE, May 1993), even though it appeared in the same journal 14 years before and does exactly the same thing!. Make sure you use Google scholar!
The Urban Transportation System Class Outline 1. Models with street grids 2. Models with street grids and arteries 3. Other models 4. Introduction to traffic congestion
The Urban Transportation System Other Forms of High-Speed Modes These methods can be applied to many other types of high-speed modes that do not go to the CBD. 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 Yinger and another from Alonso s book.
The Urban Transportation System City with Several Vertical Arteries
The Urban Transportation System Alonso: Arteries and Square Beltway
The Urban Transportation System Anas-Moses A final type of transportation system to discuss is one devised by Anas and Moses (JUE, April 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 (or, equivalently, because all people are alike, people who live in different places will make different mode choices).
The Urban Transportation System The Anas-Moses Transportation System Artery Shed Boundary Circular Street House u
