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2

tl;dr No they do not for several reasons: Your post is based on misinformation, I know of no work on optimal income taxation that would aim to 'equalize sacrifice' (heck you would not find such philosophy even among most relevant moral/political philosophies). Consequently, the whole premise is false, and you would not even expect to see arguments in favor ...

3

You question can be answered using a revealed preference argument. Let $B = \{q \in \mathbb{R}^n_+| p' q \le m\}$ be some budget set of a consumer (i.e. $B$ gives all possible bundles that the consumer can choose). Let $q^\ast$ be the optimal choice from $B$, i.e. the bundle that optimizes the utility. Then for any other bundle $q \in B$, it must be that \$u(...

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Correcting Misconceptions in the Question: Before providing an answer it is worth noting the premise in your question is simply incorrect. You state: I have heard that introducing capital gains taxes would create distortions in investments, and that the optimal tax rate for capital gains is actually 0. But this is not rationale for the famous 0 capital ...

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