The cause of the high cost of housing is not that many want to live in these cities, but rather the "rent-seeking", a market failure or market inefficiency which is caused by a politically induced scarcity of housing through the use of zoning density restrictions. The inefficient markets interfere with wealth creation and instead the use of politics transfers wealth to special interest groups, in this case wealthy landlords like President Trump from people who rent and are buying housing.
Economist David Ricardo first developed the concept in the mid-19th century and joined British Parliament just to overturn the rent-seeking which harmed the British economy. In his particular case it was the "Corn Laws" which taxed imported grains of all types with the result that bread and basic food cost more for consumers. Farmers would receive more money for their grain, but ultimately the landlords who leased the farmland to farmers simply charged the farmers more money as a result. Thus, there was a politically induced transfer of wealth (ultimately) from (often lower class) consumers to wealthy landowners.
This is analogous to what is happening in the housing market today.
Japan fixed the problem in 2002 when it passed a federal law taking zoning authority away from local municipalities. The result: in 2014 Tokyo built 140,000 housing units compared with 20,000 for all of NYC and less than 90,000 for the entire state of California.
Interested readers should consult writers of Harvard Economist Edward Glaeser, Economist and Financial Times Columnist Tim Harford (book, "The Undercover Economist"), and articles by Steve Beyer
Edward Glaeser: http://www.nydailynews.com/opinion/build-big-bill-article-1....
Tokyo's Affordable Housing Strategy: Build, Build, Build https://www.forbes.com/sites/scottbeyer/2016/08/12/tokyos-af...
What Liberals Don't Get About Affordable Housing: Filtering. https://www.forbes.com/sites/scottbeyer/2016/02/19/what-libe...
See also "Green Belt"