Organizers and their supporters routinely neglect what economists call “opportunity costs” — in this case, what might have happened if a country didn’t host the Games. In some of the world’s most expensive cities, perhaps the greatest opportunity cost is the loss of scarce and valuable real estate. While many facilities remain in use after the Games or are converted for new purposes, quite a few sit virtually as empty as the original in Olympia, Greece. Tourists can ride a Segway around the Bird’s Nest in Beijing for $20.

Similarly, it’s misleading to calculate how much money is spent in a city during the Olympics. A fair comparison requires some estimate of how much would have been spent without them. When the Games come, after all, other kinds of tourism go. During the 2012 Games, the Adelphi Theatre in London’s West End suspended performances of “Sweeney Todd.” The British Museum received 480,000 visitors that August, down from 617,000 the previous year. Indeed, Britain received about 5 percent fewer foreign visitors in August 2012 than it did in the same month the previous year. Those who showed up spent more, sure, but London spent billions of dollars to lure them. “If Boston hosts the 2024 Olympics, there’s no doubt that [the city] is going to be overrun with sports tourists,” said Victor Matheson, an economist at the College of the Holy Cross in Massachusetts. “But Boston is already overrun with tourists in the summer.”

Many hosts, of course, don’t care all that much about breaking even. The Olympics have always been a debutante’s ball for emerging economies, from Japan in 1964 and Germany in 1972 to China in 2008 and Russia in February. And there is some evidence that it works. Countries that host the Olympics experience a significant increase in trade, according to a 2009 study by Andrew K. Rose, an economist at the University of California, Berkeley, and Mark M. Spiegel, an economist at the Federal Reserve Bank of San Francisco. But their research determined that this was also true of countries that made losing bids for the Olympics — spending tens of millions rather than billions. The benefit, in other words, came from the signal that a country was open for business, not from the spending itself. “One city thinks winning the Games is worth more than all the other cities do,” said Andrew Zimbalist, an economics professor at Smith College and the author of the coming “Circus Maximus: The Economics of Hosting the Olympics and the World Cup.” “And that city is likely to be making an error.”

And while Brazil may be eager to signal its economic might, the Games can also tarnish a host country’s reputation. The enduring image of the Munich Olympics is a man in a ski mask; decaying venues from the 2004 Olympics became a metaphor for Greece’s economic crisis. Sochi’s legacy was overshadowed by security concerns and warm weather — even before Russia ended hopes for a tourism boom by annexing Crimea. In a study of the impact of the 2000 Olympics in Sydney, Australian researchers interviewed people in four countries — Hong Kong, Malaysia, South Africa and the United States — one year before and after the Games. They found little change in perceptions, with one surprising wrinkle: South Africans had soured on Australia “because of the way in which the Aboriginal issue was highlighted and portrayed by the South African media,” which drew comparisons with that nation’s history of apartheid.