[Based loosely on a problem in the notes of Emily Moore.]
In the midst of a heated debate on fixed-rate taxation, a senator proposes a sliding scale tax rate, in which your tax rate is based on one five-thousandth of your income. For example, if you make $5,000, you pay 1% of your income in taxes, if you make $10,000, you pay 2% of your income; if you make $100,000, you pay 20%; if you make $500,000 or more, you pay 100% of your income.
Obviously, this isn't the most sensible policy (unless you believe in a form of social equity that limits the top amount someone can earn). Nonetheless, we will treat it as such. Under this policy, what is the optimal income?