That’s a damn twisted idea of 'fair." The rich guy has at all times the freedom to live as rich. That’s the power of money in the bank. So don’t pretend that his ‘self-denial’ is somehow making him equal to the poor.
In fact, your hypothetical miser who doesn’t spend his money is shrinking the GDP by not spending money, which tends to depress demand relative to his bank account’s proportion of the economy. Technically, if we want to maintain sufficient economic liquidity, he should be taxed more to keep the money flowing through the economy, rather than letting some “simple life” weirdos deflate GDP.
So there. 
This.
No, wait, this makes a certain orcish sense. From another thread:
The libertarian right apparently no longer see government as charged with ruling or governing the people. The welfare state has convinced them that the government is some kind of charitable fund, & they want to opt out of it like it’s the United Way or something.
Yet when you tell them it’s the government, they call you Lenin. 