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This is way oversimplified.
I would divide it into four classes of cities.
Legacy, prosperous cities:
New York City
Chicago
San Francisco
Boston
Washington DC
Philadelphia
etc.
Legacy, decaying cities:
Cleveland
Baltimore
St Louis
New Orleans
etc
New, urban cities:
Los Angeles
Miami
San Francisco
Denver
Salt Lake City
Dallas
Houston
Atlanta
etc.
New, suburban cities:
Nashville
Charlotte
Austin
Raleigh
Phoenix
Las Vegas
etc
While I think that this is an improvement on the OP's binary categorization, I think that it's still too simplistic. I say this because there are people who are living very suburban lives around legacy cities and people who are living in the urban areas of metros that are overweighted with suburbia. Many "New, suburban cities" have old cores that date back as far as the 18th century.
I would think that there are people who might choose to live in a new suburban areas around old legacy cities and people who choose to live in the older cores of the more recent "suburban" boom towns.
I don't understand separating Denver from the newest tier. By the 30s, Houston surpassed Denver and Denver wasn't far ahead of Dallas, which then surpassed it by the 50s. Miami grew behind both Houston and Dallas.
Seattle was surpassed by Houston before the 40s and Dallas in the 60s
These "rising" cities aren't cheap, lol. Houston, Atlanta, and ESPECIALLY Miami are not cheap places to live. Houston and Atlanta are about as expensive as Chicago. Miami may be slightly more.
Legacy city is not a term used to describe all older cities. It's a term used to describe older cities which went through a period of population/employment decline in the mid to late 20th century that they have yet to recover from. Basically it includes the entire rust belt, but is more expansive than that, also including some cities in the south (e.g. Birmingham, New Orleans) and Northeast (e.g Providence, Newark, Scranton).
These "rising" cities aren't cheap, lol. Houston, Atlanta, and ESPECIALLY Miami are not cheap places to live. Houston and Atlanta are about as expensive as Chicago. Miami may be slightly more.
I think cities can be divided into essentially three groups.
Mature cities are cities which have fixed municipal boundaries and which experience negative domestic migration. However, the total population continues to grow because of both natural population growth as well as international immigration. In some cases they have been transformed heavily by gentrification, but almost always the level of young professionals moving in is less than the outflow of domestic migrants outwards.
Legacy cities have fixed borders and negative domestic migration, but they are also either stagnant or declining in terms of population. This is due to a combination of relatively little immigration (virtually every legacy city has relatively small Latino and Asian populations) along with lower levels of gentrification. Legacy cities, if they economically revitalize enough, can become mature cities (see DC).
Sunbelt cities have mostly open borders - either undeveloped land within city limits waiting for new subdivisions, or the ability to annex new land for development. As a result, they can continue to see pretty strong growth just from suburban growth. Over time, sunbelt cities can turn into mature cities if their borders are essentially closed off by incorporated suburbs (this happened to Miami and Atlanta).
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