There is no reason your currency needs to have any value whatsoever as you've described it. The exchange rate will go to zero because it doesn't reflect how people value things when making economic exchanges. They just don't value labor or existence, so any proxies for those things will go to a value of approximately zero unless you use means to keep it higher.
Written by Rex Kerr
One who rejoices when everything is made as simple as possible, but no simpler. Sayer of things that may be wrong, but not so bad that they're not even wrong.