Theorem: With constant, positive income mobility such that any income group can transition to any other income group with positive probability, a non-zero sum increase in income inequality results in an increase in expected lifetime incomes.
Proof: Expected lifetime income is the expectation of income in each income group over a constant distribution of transition probabilities. The non-zero sum increase means some groups get richer while others stagnate or get richer, but to a lesser extent. So increasing one of the summands increases the sum. QED.
Corollary: If utility is monotone in expected lifetime income (or present value of income or present consumption with access to capital markets) and at least weak gains in income are experienced in every income group, income inequality doesn’t matter; mobility does.
I’ve ignored discounting which I think is what plays a part in the politics of redistribution. Suppose I expect to transition to the very richest income category sometime late in my life. If I discount the future enough, I won’t be able to wait to become rich and I may call for redistribution today.