JUST seven years after the biggest housing bubble in American history began to deflate, could another be inflating? Prices in a 20-city index compiled by CoreLogic Case-Shiller rose by 11% in the year to the end of March, and by more than 20% in Phoenix and Las Vegas, both cities at the centre of the housing collapse. Inventory is down: homes are selling in days, and often for more than the asking price. In Phoenix, bidding wars have broken out between would-be homeowners and investors paying cash. Americans once more see property as a winning asset. But to qualify as a bubble, an asset must not simply appreciate; it must decouple from its intrinsic value. For houses, The Economist each quarter compares the ratio of prices to household income and rents against their long-run average in 20 countries. We have now done the same for the 20 metropolitan areas in the Case-Shiller index. The verdict: in most markets houses are at or near their long-run values, but none looks bubbly (see article).

For America as a whole and in most cities, price-to-rent and price-to-income ratios are at or near their 25-year average. (To be sure, the bubble era dragged that average up; valuations are still higher than in the 1990s.) How they got there varies, however. Cities in Arizona, California, Nevada and Florida experienced the biggest bubbles, and in the subsequent bust values fell well below long-term averages. Price rises in Phoenix, Tampa and Miami have restored values only to their long-run averages.

In New York prices never gave up much of their bubble-era rise, and have since recovered more slowly than in the country as a whole. Relative to rents and incomes, valuations have been flat or down slightly. Homes around Washington, DC sell for roughly double their level of the late 1990s, but that seems justified by strong gains in rents and incomes. A brisk rental market also explains the strong run-up in prices in San Francisco.

Atlanta, Dallas, Cleveland and Detroit never had bubbles to speak of, and not much of a bust. Valuations in all these cities are at or below their long-run average. Chicago’s bubble was also relatively modest; valuations there still look cheap.