The Gini coefficient is a popular tool used by economists to measure economic inequality. Unfortunately, economists tend to rely on income data, which is not a very good measure of economic well being. Conceptually we'd like to use consumption data, but that is more difficult to measure.
In Real Wage Inequality (PDF), author Enrico Moretti looks at cost-of-living data in U.S. cities from 1980 to 2000 in order to determine whether growing nominal wage inequality is really indicative of a growing disparity in living standards. Because college-educated workers are more likely to live in cities with higher costs of living, especially higher housing costs, some of their newfound income gains must go toward paying for life in these expensive areas. The growing disparity in wages that we observe is partially an illusion.
Whether the wage inequality we see in the data translates into actual inequality in standards of living turns on a key question: are college-educated workers grudgingly relocating to expensive cities because that's where they can be the most productive and earn the most? Or is it because they really like those cities and are willing to pay more to live there?
If the former is true, high-skill jobs in sectors like finance and technology just happen to be concentrated in coastal cities that are not especially attractive places to live, but do have high housing costs. In this scenario, high-skilled workers are unlucky in where their jobs are located, and inequality is not as severe as it appears from income data alone.
If the latter is true, then living in these expensive cities is a sort of luxury good that is increasingly only affordable for the well-educated. High-skilled people choose to relocate there because they can command high incomes to afford the higher rents. Meanwhile, low-skilled people living in these cities face rising rents and may have to move away.
These possibilities are not mutually exclusive, and there is probably some truth to both stories.
I find a third hypothesis to be more plausible. I believe that residents of 2000 sq. foot condos on the Upper East Side are better off than they would be living in a 2000 sq. foot home in Kansas. And I also believe that residents of 2000 sq. foot homes in Kansas are better off than they would be living in a 2000 sq. foot condo on the Upper East Side.
Tastes differ, and people sort into areas where they prefer to live. Of course I'm staking out an extreme position, and there are obviously lots of exceptions. But that's also true of the other two hypotheses.
It would be very interesting for someone to take a data set like Zillow and compute the Gini coefficient for the value of housing units, and another one for the size of housing units. The value of house Gini would probably show much less inequality than income Ginis, and the size of house Gini would show even less inequality. Unfortunately, this only deals with owner occupied units, so another technique would be needed for apartments.
I also believe that the value of house Gini would be a decent proxy for nominal consumption inequality, as consumption is often roughly proportional to housing costs. In one sense it would underestimate housing inequality----richer people often own multiple homes. But in another sense it would overestimate housing inequality, as "life cycle changes" result in measured inequality even in a society where everyone has the same lifetime consumption. I.e., I consume much more housing than when I was 20 years old. My hunch is that these two biases roughly offset, leaving value of housing unit inequality a pretty good proxy for nominal consumption inequality.
And I'd say the same about the Gini for housing size. I'd guess this is much lower (i.e. more equal) than the Gini for housing values, and I'd also guess that this reflects the fact that real consumption inequality is not as bad as nominal consumption inequality. Residents of San Francisco may consume more than residents of Kansas, but their extra consumption is much less in real terms than in nominal terms.
These are all guesses on my part, and maybe someone has already studied this type of inequality data. It's not my area of expertise. But if I did focus on inequality, housing data would interest me far more than income data.