I work for a start-up that has encountered an economics problem no one on our team has been able to crack.
A story form and a general form of the problem are below. I believe they are equivalent, but if there is a difference, please defer to the general form.
We understand there is no precise answer to this question and that the value of any currency considers prices in other factors that this question ignores. Still, we figured, or hoped, that there was some general way of approximating this, even if it's under idealized conditions. Thanks!
The sultan of Slickcrudistan has gone insane. He has decreed that his country will only accept pre-2015 sand dollars in exchange for oil1. Barrels of oil will be sold to the bidder offering the most sand dollars per barrel. Additionally, the sultan is willing to sell sand dollars back to the public at approximately the current US dollar to sand dollar exchange rate.
His country produces 20,000,000 barrels of oil at a market value of $1,000,000,000 USD/year. Additionally, 100,000,000 pre-2015 sand dollars exist. Assuming liquid and efficient markets, what is the equilibrium price of a sand dollar? How would this change if there were a stable inflation in the quantity of sand dollars?
1 Suppose an extremely cheap process exists for verifying this.
Currency X has a total circulation of a units and a stable inflation rate of currency units of m percent.
Currency Y has a total circulation of b units and a stable inflation rate of currency units of n percent.
Currency Y is completely debased and of no value in comparison to currency X (at market exchange rates, b << a ).
A market with a value of c units per year in currency X can now only be transacted in a currency Y. c is growing at a stable rate of p percent.
Assuming liquid and efficient market, what is the new equilibrium of exchange?