SINCE March of last year the Case-Shiller 20-city home-price index has risen by 21%. Some of the markets within that index have risen even more; the Los Angeles index has doubled CORRECTION: has risen 30% from its post-crisis bottom. Rising prices have many financial-market observers worried that the bubble is reinflating, helped along by low interest rates and a pipeline of mortgage credit owing to the Federal Reserve's purchases of mortgage-backed securities.

But that raises the question: just what is a bubble? Use of the term has grown so fast and loose that nearly any sustained rise in prices in any market gets the bubble label at some point. I'd say there are a few ways to think about what a bubble is or might be.

A bubble could be a pure, financial-market ponzi game, in which prices bear no relation to any defensible story about future fundamentals. It may depend entirely on buyers betting prices will go up because others will buy betting prices will go up, and so the bubble inflates until the supply of suckers is exhausted.

But a "bubble" might have a more rational basis. It could be a sustained rise in prices that is reasonable given particular, defensible expectations about the future, but which deflates once it becomes clear that particular version of the future has not come to pass. The future is uncertain, after all, and it is not irrational to bet that something might happen just because it doesn't eventually happen.

Or, a "bubble" could be a sustained rise in prices that is reasonable given particular, defensible expectations about the future—which eventually are borne out—but which nonetheless deflates, temporarily, due to an unexpected turn in credit conditions.

Now, which of these was America's housing bubble? Before you answer, it's worth going back to look at a Free exchange column written earlier this year, on an interesting working paper by Ed Glaeser, which describes America's history of real-estate booms and busts. The column examines his take on the great Chicago land boom of the early 19th century:

At the time, water access was critical to trade. In 1816 it cost as much to move goods 30 miles over land as to ship them across the Atlantic Ocean. Land near key ports and shipping routes was therefore extremely valuable. The Erie Canal led to economic booms around the Great Lakes, and Chicago’s proximity to the Mississippi river system made it an attractive bet. In 1830 Chicago land went for a song at $800 per acre (in 2012 dollars). In just six years the value soared to $327,000 per acre, with some plots fetching $1m. Tighter international credit conditions led to panic in 1837. By 1841 prices had fallen back to $38,000 per acre. Yet this was more a product of unpredictability than irrationality. Given the risk that Chicago might fail to become a great metropolis, values immediately after the crash look low but justifiable. Prices at the peak were also consistent with reasonable views. At the time Chicago’s prospects looked uniquely bright. Land values in 1836 made sense given the defensible assumption that Chicago prices would rise to a fourth of those in New York city. And, Mr Glaeser notes, people who bought and held land through the crash prospered over the next two decades: average annual returns through to 1856 were about 9%.

From the perspective of 1841 Chicago was obviously a bubble; nothing could have been clearer. And yet an investor who had purchased the most expensive tract in Chicago at the very height of the boom would still have earned a real return of 3.6% per year over the next 20 years.

As Mr Glaeser notes, there is a selection bias at work. Chicago became Chicago and so bets at the peak of the Chicago market were, ex post, sensible. Yet there were other cities that might have become Chicago but didn't, where bets at the top of the market didn't pay off.

Calculated Risk gathers the latest Case-Shiller data into this helpful chart: