The main cost is that if one economy sinks, the ones to which it is tied may sink with it. Many Germans are probably now regretting their membership in the EU, and many people in the U.K. are probably glad that Margaret Thatcher was such a strong opponent on involvement in a single currency:
Another problem that results from the issues mentioned above is that countries like Greece are held to the fire to make economic reforms that are not always in the best interests of the countries in question. The austerity packages implemented in Greece are the result of external pressures rather than any democratic agreements. The burden of repaying debt has fallen hardest on the people who are least able to pay it, especially young people. This is not to say the austerity programs are a bad idea- indeed, they are probably necessary. But the taxation reforms that accompany them are basically regressive, and the public sector is being gutted.
All of the things that are noted above seem to be right on the "money." (Sorry.) For many years we have seen how the U.S. dollar changes in relation to other countries. If the dollar is weak, it is not worth as much in other places (currency conversions). Devaluation is when currency loses its strength. Some analysts believe that when a devaluation takes place it can boost the economy by promoting more manufacturing.
If our dollar is faltering, the only places where the dollar is strong is in economies that are worse off than our own. When the dollar is strong, the economy is strong. The mention of Greece is a perfect example of the effect of a country's economy and struggling currency affecting other countries who are tied to the failing country and its currency (and its economy's difficulties). This seems simply a larger example of what we have been experiencing in our country as we have seen our government bail out banks, etc. Overall, when the currency of a country becomes unstable, people stop investing in that country. If there is already a economic crisis in that country, this will only make it worse.
These problems illustrate the "Tragedy of Commons." If all countries forgo their sovereignty by means of a common currency, the thought is that the poorer countries will benefit at the expense of the richer countries. However, this is not so -- since no country actually has a currency, economies will become destabilized. Greece has failed precisely because those who want a world currency believed Greece was "too big to fail," and that like a good neighbor, the richer countries would ( be forced to?) help out. Their reaction is predictable; they want no part of it.
I have to agree with the problems noted above regarding single currency. Another issue which may arise is that cost of living is very different for different places. For example, a dollar in one state in America will get you more than a dollar in other states. (While this is a simplistic example, it shows the problems which arise when multiple countries share the same currency). Not all places which share the same currency pay the same amounts for necessities.
The problem of a single currency when things are not going well is that the stronger countries have to bail out weaker ones. Put it this way. Why would a hard working factory worker in Germany who have been saving and living frugally have to bail out a country that has been fiscally irresponsible? This is exactly what is happening in Europe. Greece is not doing well, but neither is Spain, Portugal, and Italy. They are all part of the same currency and this will hurt all the countries that are doing well.
There are costs to the countries involved as we are now seeing. If Greece was not in the euro zone, it could devalue its currency and help to get out of its problems in that way. The countries that are doing better are seeing costs as well. If Germany weren't tied to countries like Greece and Portugal, they would not have to worry about having to support those countries to prevent a major decline in the value of their common currency. Germany would be able to let those countries go their own way since they would have separate currencies.