For the sake of argument, let's suppose the total value of the coins on the island is $210.00 which comes out to 300 cents per person. I would propose we start using that as the island's money.
So we do that - now where does the "economics" theory take place?
This would allow the inhabitants to exchange their goods and services with each other fairly. Some would learn to fish, some would collect rain water, some would farm, some would build dwellings, others would collect firewood. People on the island would be able to change their "professions" based on the supply of and the demand for the various products and services.
The biggest difference between Austrians and Keynesians is related to the supply of money. I believe a fixed supply is the only fair way to allocate limited resources. If the island's inhabitants were to designate a few individuals to be bankers to control the supply of money, then they would be able to create more money, allowing them to consume products and services that others produce and provide while contributing nothing of value to others on the island.
Indeed, the problem with money creation is the seigniorage. It is btw what we are all after if we hold bitcoin :-)
Seigniorage is unfair because it is value obtained for nothing (no production, and no investment or risk-taking in any production, no saving).
If you increase the money supply, there is of course seigniorage, and that is paid for by those already holding the money, but who are not allowed to make more of it. The creators of the money supply tax those that hold the money (and money is always held !).
Keynes made the fundamental error in thinking that the price of MONEY is the interest rate. In fact, interest rate is the price of *holding value*. Keynes' basic idea was that one can get the interest rate lower by increasing the money supply.
If you increase the money supply, you create inflation. It means that to hold the same amount of value, you will need to hold MORE money. So if the demand for holding value is the same, increasing the money supply to alleviate this won't help, because the inflation will make this correspond to holding MORE money.
You can see this as follows:
Suppose the market wants to hold 1/3 of the money market cap of value Q, and that this leads to an interest rate of 15% say.
Now you want to relieve this. You print twice as much money (inflation a factor of 2). Point is, the market cap in VALUE is still Q, but it corresponds to twice as much money. If the market still wants to hold 1/3 of the market cap Q, this will now correspond to twice the amount of money that was demanded before. So you've printed twice as much, but the demand is twice as much. Keynesian printing shouldn't affect interest rates.
In reality that doesn't happen. Keynesian printing does lower interest rates. In fact, what happens is that if you start printing, that those wanting to hold value ESCAPE into other assets than your inflating money. So yes, the DEMAND for money will lower, the interest rates will lower.... and you are inflating the demand for *other* stores of value (houses, dot com stock....). You won't see this. You are blowing a bubble somewhere else without noticing. This initial rising first has positive effects on many indicators. People start thinking that the economy takes off. And then the bubble bursts. And Keynesians are going to print even more to try to erase the damage from the burst bubble. To blow another one. And so on.
In the mean time, a lot of seigniorage has been produced, and those that are profiting from it get richer and richer.