Bill Easterly makes a joke about trying to measure the impact of participating in the Olympics influence on winning medals. For countries who didn’t earn medals in London a study would show participating has no influence on medals earned. At the core of the joke is you should carefully choose what you are measuring. International aid programs might increase school enrollment or the number of businesses but it might not increase income or the number of PhD earned.
What is the goal of countries like Chile who didn’t win any medals at the Olympics? Was it a complete failure if a country goes home without any medals? I don’t think the Jamaican bobsled team (made famous by the movie Cool Runnings) would consider their participation and last place finish in the 1988 Olympics a failure. Even the bobsled team used that event to build toward eventually having a Jamaican win a bobsled medal at the 2006 Olympics. So perhaps participating with little chance of winning increases future chances of medals.
The next question is what is the value to a country of an Olympic medal? For a smaller country like say Costa Rica, there was a lot of national pride gained by Claudia Poll’s 1996 Gold medal and when I was there in 2000 the country was very excited to watch her swim again. I think an economic value could be put on this pride. Typically economists would use a method called Contingent Valuation where they survey people how much is a medal for your country worth to you or even ask people to donate or pay taxes to a national Olympic committee. This paper by Brad Humphreys and others suggests Canada's big investment in 2010 paid off 2-3 times the cost based on surveying people's value of Olympic medals.
It is unclear how much a country like Chile or Costa Rica should invest in Olympians, but economic tools can help find the answer. My guess is that the answer is not zero, unless like the United States the market takes care of this investment through sponsorships.