Let's be honest; if Atlanta, on its own, dumped billions into revitalizing itself, no one would care except residents. Having a stage like the olympics to draw the world in, however, shows people the city and makes it more likely they will return.
Likewise with Vancouver; one of our major downtown streets, Granville, got a complete overhaul. Gone were the old, broken stones, the cracked sidewalks, the run-down streets and sickly trees. It's now a much better place to walk around and explore.
We also have a new mass-transit line connecting our downtown core with South Vancouver and Richmond, as well as the airport. Catching a flight used to be $25-35 by taxi, or god knows how much in parking. Now, it's about $4 per person, and that gets you there and back if you're dropping someone off. The project was a public-private partnership that was within budget and opened fifteen weeks early, and has made Richmond far easier to travel to and from (making everyone's life easier). The project was a no-brainer improvement to the region, but I doubt we would have had as much support and funding if we hadn't had the olympics coming to town.
Whether the other renovations to the city will prove useful, it's hard to tell yet, but I'm cautiously optimistic that they will.
Olympics are rare enough that I don't know if any formal study has been done.
However, similar arguments are made all the time in support of building new sports stadiums. And at least in those cases, the arguments simply don't hold water: the beneficiary is the team ownership, NOT the taxpayers.
See http://www.cato.org/pubs/pas/pa-339es.html :
The economic facts, however, do not support the position that professional sports teams should receive taxpayer subsidies. The lone beneficiaries of sports subsidies are team owners and players. ... Indeed, the results of studies on changes in the economy resulting from the presence of stadiums, arenas, and sports teams show no positive economic impact from professional sports -- or a possible negative effect.
Kind of aside the main discussion, but the Canada line was approved at 1.35 billion, cost 1.9 billion, and had it's scope significantly reduced in order to manage that. It being a p3 allowed hiding 'competitive' information such as cheaper but disruptive construction and single track sections from the public, and the cities involved until after it was approved. I like having new mass transit, but I'm not sure this can be used as an argument for the economic benefits of the olympics. Though it could be an argument that the olympics give political cover to push through large projects irrespective of the problems/benefits and that may or may not come out well.
I don't know if you can say the new line made commuting around Vancouver any easier—more modernized, perhaps, but until it opened there were bus lines running down the same "track" with the same carrying capacities/latencies (98s and 424s.) They also removed many long-distance "shortcut" buses in the process, meaning that getting from, say, 1 Rd. in Richmond to Granville & Broadway turned from one bus to three (2 + the train line.) It was definitely extra capacity required for the Olympic duration, but it's mostly a wash now that it's over.
Other ways to spend the money? Well, anyone living in Atlanta now will tell you this winter's rainfall showed them just how bad the city's sewer system is. Its well known that Atlanta has major sewer problems for perhaps 100 years...think "massive sink holes that can suck down an entire building".
When your spending on beautification to show the world your city for 3 weeks, major infrastructure projects like sewer systems aren't high on the list.
