THE city that proved that America’s housing market is rising from the ashes was, fittingly, Phoenix. From property-bubble peak to post-financial crisis trough, house prices in Arizona’s biggest metropolis plunged by more than half, as demand dried up and foreclosures soared. But low interest rates and slowly reviving consumer confidence, plus a bunch of private-equity firms convinced there was an opportunity in buying up homes cheaply, eventually brought the market back to life. By September the median sale price of a single-family home in the Greater Phoenix area was $199,000, up 33% in a year.

Yet lately not all the signs from Phoenix have been positive. Initially, house prices and sales rose together. But according to a recent report by the Centre for Real Estate Theory and Practice at Arizona State University, they have started to diverge. In September the number of single-family homes sold was 9% lower than in the same month a year earlier; sales of town houses and condos were flat. Since July the Greater Phoenix market has “cooled dramatically”, thanks to a steep fall in demand. In 2014, the report predicted, prices will rise more slowly than the “furious pace we have witnessed over the past two years.”

A similar picture is emerging elsewhere, especially in cities where price and sales growth had been strong. The monthly index of housing conditions published by the National Association of Home Builders and Wells Fargo, a bank, published on November 18th, was flat since October and down from 59 in July to 54 now. (That is slightly positive; a score of 50 would mean that builders are evenly divided as to whether conditions are good or bad). The Case-Shiller index, which tracks house prices in 20 big cities across America, reported year-on-year price growth of 12.8% in August, but also noted an abrupt slowing of the monthly rate of growth as 16 of those cities reported more modest price increases in August than in July. In October the Federal Reserve noted that the “recovery in the housing sector slowed somewhat in recent months.” This set the stage for the release of data on November 20th that showed a sharp 3.2% fall in sales of existing homes in October.

This is probably evidence of “stalling, rather than a reversal”, says Jim O’Sullivan of High Frequency Economics, a research firm. Would-be buyers are coming to terms with costlier mortgages; long-term mortgage-interest rates are up by a percentage point since the early summer. That can be a drain on a tight household budget. The housing-affordability index published by the National Association of Realtors, which combines average mortgage costs, average home prices and average family income, is down to 164 from a high of 214 in January, which is significant. Yet housing remains far more affordable than the historic average of around 125, which means that the recovery is likely to pick up before long, says Mr O’Sullivan.

What effect will this slowdown will have on builders? Previous busts have taught them to control the inventory of new properties coming to the market. If they react abruptly to falling sales by building less, the housing market may be “on the verge of a significant correction”, argues Ian Shepherdson of Pantheon Macroeconomics. Although residential construction is only 3% of GDP, in each of the past five quarters it has contributed around a quarter of America’s economic growth.

If residential construction turns down, asks Mr Shepherdson, where else will growth come from? Corporate America seems to be hunkering down and curbing capital spending. Exports? Maybe. Higher consumption? Probably not while housing is losing momentum. Government spending seems unlikely to come to the rescue, even if the politicians behave themselves. With luck, signs of a weaker housing market will concentrate minds in Washington, DC as the next decision over the debt ceiling approaches.