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Some startling math from the country's most expensive cities.

High-cost cities tend to have higher median incomes, which leads to the simple heuristic that, sure, it's costlier to live in San Francisco than in Akron, but the people who pay bills there make enough money that they can afford it.

In reality, yes, the median household income in metropolitan San Francisco is higher than it is in Akron (by about $30,000). But that smaller income will buy you much, much more in Ohio. To be more specific, if you make the median income in Akron – a good proxy for a spot in the local middle class – 86 percent of the homes on the market there this month are likely within your budget.

If you're middle-class in San Francisco, on the other hand, that figure is just 14 percent. Your money will buy you no more than 1,000 square feet on average. That property likely isn't located where you'd like to live. And the options available to you on the market are even fewer than they were just a year ago, according to data crunched by Trulia. To frame this another way, the median income in metro San Francisco is about 60 percent higher than it is in Akron. But the median for-sale housing price per square foot today is about 700 percent higher.

The gulf between those two numbers means that the most expensive U.S. cities aren't just unaffordable for the average American middle-class family; they're unaffordable to the relatively well-off middle class by local standards, too.

To use an even more extreme example, the median income in metropolitan New York is about $56,000 (including families in the surrounding suburbs). If someone making that much money wanted to buy a home on the market this October in Manhattan, the most expensive home they could afford would cost about $274,000. A mere 2.5 percent of for-sale housing that's available in Manhattan now costs that little. Oh – and those properties are averaging 500 square feet.

Trulia ran these numbers based on the assumption that a family shouldn't spend more than 31 percent of its pre-tax income on housing (and that it must pay local property taxes and insurance). This data also assumes that a family makes a 20 percent down payment on a home – a daunting feat even on a six-figure income in somewhere like Los Angeles or New York.

By those calculations, these 10 metros are the least affordable, using Census data on median incomes (note that the data refers to metros, not cities):

Data courtesy of Trulia

In San Francisco, a household making $78,840 a year can top out buying a home worth about $409,000. 24 percent of the homes for sale in the area were below that threshold last October. Now it's just 14 percent. In fact, in every one of those 10 metros, a smaller share of homes are considered affordable now to the middle class than last year.

The same trend is true even in those metros where the vast majority of housing is accessible on a local median income:

Data courtesy of Trulia

Affordability is effectively declining as home prices are rising (and at a much faster rate than median incomes). Within the most expensive metros, the most affordable housing is also located in the areas that require some of the longest commutes. In metro New York, for instance, the Bronx and Nassau County are home to the bulk of the most affordable housing in the region.

Or, there's always a move to Akron. Here is the full data from the 100 metros that Trulia examined:

 

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