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Density Functions

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  1. Density Functions Chapters 8, 10 (c) Allen C. Goodman, 2006

  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 A> 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 isn’t 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. • There’s 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. International Perspective From Alain Bertaud, 2003 Figure 1:Three dimensional views of population distributions in 7 cities represented at the same scale

  18. Built-up Densities around the world (figure 2) 640 acres/sq.mile 259 hectares/ sm Bombay Approx 101,000/sq.mile

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