Are high density cities more exspensive to live in than lower ones?

Comparing San Fransico to say Phoenix or Dallas

Yes and no. Generally speaking, anything you want to have will cost you more in a high-density city. At the same time, many things that are considered mandatory in a low-density city (from a back yard to a car) are entirely optional in a high-density city...
Generally yes. I can't compare the cities you list, but generally the answer is yes. You would think that more people means more businesses means more competition means lower prices. But you said it already, high density, means high demand for real estate, means expensive land costs, means costs of doing any business is higher. The thing is San Fransisco being located where it is has waterfront property which is always more expensive as you pay for the view. And it has limited land to expand on. That verses Phoenix that may have room to expand out being in the desert.
Yes high density cities are more expensive; just the housing can give you a hint. NY, LA, S F, are good examples.
I tried to think about the cities I am familiar with outside of the U.S., to see if density really does have a direct relationship to cost of living (living expenses), as you suggest. Definitely Singapore, Hong Kong, London, Paris, New York, Moscow and Tokyo are all very expensive cities to live in. (Moscow was the most expensive city in the world in 2005 and 2006!) Surprisingly, I couldn't really think of any low or lower density cities outside of the U.S. I think that places like Phoenix and Dallas may be a U.S. phenomenon because we do have the space/room to accommodate sprawl in this country.

So, I'd say yes to your question, which was a really good one.

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