I'm an engineer and my math is pretty rusty so please bear with me.
We need a function that approaches infinity at a given bound (however the bound is an input to the function, so basic polynomial interpolation did not work for us). We think this would be a variant of a vertical asymptote function.
The purpose of the function will be to calculate a price which increases similar to a linear function at first, but then as supply reaches a bound, it goes to infinity.
We came up with something that is close:
$p = (d-L-3)/(d-L) -3/L -i -1$
p : current price
d : demand (units in circulation)
L : supply limit threshold
i : initial price
For example, here is a graph with L=50 and i=10:
The problem is it is too "flat" or "steep" for different input values. We want the price to double midway to the bound and triple at about 75%.
We tried adjusting constants, even if we get it correct for a specific (i,L) input value pair, it is incorrect for other input values.