Density Functions

1
Density Functions

• Chapters 8, 10

2
Rent Functions

• We saw that the competitive bidding for land
  yielded rent curves that looked like this

Rent
Distance
3
Impacts of Decreasing Rent

• We substitute capital for land, where land rent
  is high ? vertical city.
• We substitute land for capital, where land rent
  is low ? horizontal city.
• What happens over time?

4
Over Time

• Agt Remember ?p -t/h. Over time, out-of-pocket
  t ?, making the numerator small. Why?
• However, as income ?, valuation of time may rise,
  so technically, travel costs could go either way.
• See time_costs.xls

5
Over Time

• Remember ?p -t/h
• Over time, income ? making housing ?, making the
  denominator large. Why?
• What happens to rent function?

6
Over Time

• Consumers bids (per mile) tend to get shallower.
  Locating near the center isnt important. Lots
  of urban analysts call this the traditional
  model of decentralization.
• How do we measure it?

7
Density
Do

• Density Density at the center, multiplied by a
  decaying factor.
• D (u) Do exp (-?u), or
• D (u) Do e- ?u
• The larger the value of ?, the steeper the
  density function, the more centralized.

Density
Distance
8
Density

• Over time, we would expect that urban areas
  become less dense, so the function becomes
  flatter.
• Theres a whole host of empirical work that bears
  this out for population, housing, employment.

Early
Density
Later
Distance
9
Decentralization

• Clearly, the population has decentralized.
  Percentages of populations in central cities have
  fallen continuously for as long as we can
  measure.
• This has happened for both employment and for
  residences.

10
McMillen Finds
McMillen, Daniel P., Polycentric urban
structure The case of Milwaukee, Federal
Reserve Bank of Chicago, Economic Perspectives,
15-27
11
Transportation Changes
Land rent

• Key changes -- Ability to move goods on roads.
  Intracity and intercity trucks.
• You can move goods out from central shipping
  point.
• You can ship directly without going to center

Bid function w/ horse-drawn wagon
Bid function w/ truck
Residential
Distance
12
So, Why Suburbanization?

• Increase in real income
• Decrease in commuting cost
• Central-city problems race, crime, taxes,
  education
• Following firms to the suburbs
• Public policy

13
Public Policy?

• Subsidies for home-ownership
• Commuting externalities
• Fragmented system of local government has
  suburbs competing with central cities.
• Highway construction

14
Gentrification?

• Coming back to the city?
• Who does it?
• Wealthy, young, highly educated
• Relatively high commuting costs
• Relatively low demands for housing and land
• Few children
• Most move in from elsewhere within city
• Many move out when children get older

15
Subcenters Los Angeles

• Come from agglomeration economies.
• Important for employment and commuting.
• CBD is still largest.

16
Density with Subcenters
Density

• City and metropolitan area may have bumps.

Distance
17
Figure 1 Three dimensional views of
population distributions in 7 cities represented
at the same scale
International Perspective From Alain Bertaud,
2003
18
Built-up Densities around the world (figure 2)
640 acres/sq.mile 259 hectares/ sm
Bombay Approx 101,000/sq.mile