Could a country (or economic union) run entirely on a currency that's completely digital? And by that, I don't mean that it can be destroyed if the servers are shut down (like a cryptocurrency), but that there are no physical denominations of the currency in circulation (much like a credit or debit card).
Such a currency exists already - bitcoin.
In any digital currency if ALL servers running related programs/containing related files are shut down the currency will stop working, but if that nation adds a law that all computing systems with more than # processing power must contribute to the bitcoin (or whatever currency you want to use) network that should really not be a possible problem. Such a law is not unrealistic either, given the many other things we are forced to do by law (pay taxes, buy insurances etc.). The currency could easily be incorporated into the operating system and the user wouldn't even notice.
By definition it isn't possible for physical bitcoins to exist and plenty of systems are already in place to easily pay with bitcoin instead of physical money if the seller supports it, in a theoretical world where bitcoin is the only currency (even if it's only in a specific nation) every seller would have to support it or give away their wares for free (I'll let you guess which option they'd choose).
Since we're on a site for building fictional worlds, it is clearly possible. There would be problems, but those problems can be overcome, or they can become part of the story/adventure.
The physical coins and bills are just tokens, their intrinsic worth is not related to their face value. Large-denomination coins and bills tend to be worth more than their production costs, small-denomination coins less. There are no fundamental difference between metal, paper, plastic, or digital tokens in this regard.
Physical tokens have advantages for certain transactions.
- I can give them as a gift with symbolic and monetary value. Giving or receiving a crisp €100 bill feels different than watching my account go from €1,352.48 to €1,452.48.
- I can have small, untraceable, quick transactions. I can give a beggar 50 cents without giving him any data about me. I can decide to spend 90 cents on a donut or 83 cents on an apple, and nobody has to know if I made the healthy choice or not.
- I can give or receive them without technical systems. When people forget their wallet or have it stolen, they might be able to borrow some cash from friends or colleagues.
- I can keep track of the sum of little transactions. I tend to get the same amount from an ATM whenever my wallet looks empty, and if I do that twice in a week I briefly think about what I'm doing and if there were any extraordinary expenses. I would not remember cent-level digital transactions that way until I check the bank statements.
Physical tokens have disadvantages for certain transactions.
- Carrying large amounts of cash makes me uneasy. What if I forget my wallet? What if it is stolen? A secure digital system would help.
- These days I get the physical tokens at an ATM and spend them at various shops. Going to the ATM on the way to or from work requires a slight detour. No problem on most days, but I often find that I've automatically taken the direct route. I could save the detour if everything was digital.
- A digital system might be easier to audit. As a conscientious taxpayer, I approve of systems that make tax fraud or money laundering harder. (But see the privacy concerns, above.)
Yes, such a system could exist, but...
With a fully digitized currency, some form of physical trading system would likely develop for illicit or 'off book' purchases. Gold and silver are obvious choices to be used. Or foreign currencies, if they still exist. Or perhaps something less obvious, like the pouches pouches of tuna used in some prisons. Whatever it is, there would be some sort of physical trading medium in place.
Look at bitcoin, for example. While its the opposite (an all-electronic option to physical currency), it demonstrates that alternatives develop where there is a void.
It would be possible if they had some sort of device which they could carry around everywhere which transferred the money between people. Something like a smartphone could be used. The hard thing would get people to switch between physical money and digital. It would be impossible to "print" money because it is all digital so the economy would be much more stable if the government or whatever organisation is running the economy had very tight security. Taxes would be automatically deducted which would help with people not paying taxes and it seems very likely that this sort of system might happen in the future.