But it is still up for debate whether they should or not. Taking the other side is Nobel-prize economist Paul Krugman. He makes his opinion crystal clear:
"So here it is: yes, Cyprus should leave the euro. Now." New York Times 3/26/2013
Krugman is coming from an academia economics perspective, where the literature on the topic is clear about the costs and benefits of sovereign defaults. Reinhart and Rogoff prove that countries are better off once they clear themselves of debt (also that countries keep getting themselves into the same messes using the same poor rationalizations of their problems). The market does not punish countries for defaulting. Krugman adds to this by showing that Iceland is making a swifter recovery by defaulting on its debt, then debasing its currency. Poland was able to continue despite the global recession because they debased their currency.
Most currencies are free-float on the open market. Value is determined by supply and demand. Many factors determine demand, including interest rates, economic health, foreign trade. While Supply is determined by the Central Bank. Pre-Eurozone Europe, the Northern countries (e.g. Deutsche mark and the French Franc) had stronger economies (growth, inflation, debt burden) therefore stronger currencies relative to the South (e.g. Spanish peseta and the Greek drachma).
Now in the Eurozone, all the qualities of these economies are bundled in one currency. The Eurozone has a single monetary policy for 17 very different economies going through very different problems. A one-size fits all solution is not working. The current Euro is weaker than a Northern Euro would be, while it is stronger than a Southern Euro would be.
Krugman is a proponent of the smaller, weaker countries in the Eurozone leaving for their own good. Countries like Greece, Cyprus, and Slovenia (maybe even Ireland) could go through a managed bankruptcy, default on their debt, leave the Euro, and be better off for it in the long run. Their new currencies would be free-floating and therefore weaker, which will aid in their economic recoveries by making it cheaper for other countries to purchase their products and services.
The fear is that a smaller, weaker country leaving could have a destabilizing effect on the larger, stronger economies of the North. If Cyprus completely defaults, then it might destabilize the Spanish banking system, which could then domino into Italy, France, and Germany. You better believe that the Eurozone would economically retaliate if any of its members were to jump ship, and therefore knock the first domino down. This is what Krugman does not take into account, because there is no economic literature on such a unique political science problem.
It is also possible (though less likely) that if the smaller, weaker economies leave the Euro, then the rest of the Eurozone could emerge stronger for it. A survival of the fittest makes the whole herd stronger theory. This is what Krugman believes, but it would have to be done very, very carefully. The current leaders of the Eurozone believe the downside risks of pursuing this course of action outweigh the possible benefits. I agree with them.