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

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

Rent Functions We saw that the competitive bidding for land yielded rent curves that looked like this: Distance Rent

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?

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.xlstime_costs.xls

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

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?

Density Density = Density at the center, multiplied by a “decaying” factor. D (u) = D o exp (-  u), or D (u) = D o e -  u The larger the value of , the steeper the density function, the more centralized. Distance DoDo 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. Distance Early Later Density

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.

McMillen Finds: McMillen, Daniel P., “Polycentric urban structure: The case of Milwaukee,” Federal Reserve Bank of Chicago, Economic Perspectives, 15-27

Transportation Changes 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 Distance Land rent Bid function w/ horse-drawn wagon Bid function w/ truck Residential

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

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

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

Subcenters – Los Angeles Come from agglomeration economies. Important for employment and commuting. CBD is still largest.

Density with Subcenters City and metropolitan area may have “bumps.” Distance Density

Figure 1: Three dimensional views of population distributions in 7 cities represented at the same scale International Perspective From Alain Bertaud, 2003

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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