Oops, one additional point on equal sacrifice theory and classical benefits-based taxation

I see that in my Part 1 post on the Weinzierl paper from yesterday's colloquium I promised to mention something at the end of my Part 2 post, and then I never did.  So I'll add that final point here.

Specifically, I noted that "Mill, in proposing equal sacrifice theory as a replacement for Smith's benefit theory, not only reached a very similar conclusion (flat tax above an exemption for necessities), but cited Smith as an influence and precursor rather than as the dead past that he wanted to bury."  I then offered to make an observation about this overlap.

One interpretation would be that equal sacrifice theory and benefits theory lead to a similar place, e.g., they might plausibly support a flat rate tax.  But, with all due respect for Smith and Mill, I question this because the underlying elasticities on which they turn are both (a) entirely distinct and (b) extremely hard to tease out - not just empirically but as a basic conceptual matter.

Again, under equal sacrifice the key elasticity that drives the optimal rate structure pertains to utility loss per dollar of tax paid as income rises.  This is against the background of a counter-factual hypothetical in which public goods were somehow provided for free.  

Under classical benefits-based taxation, the key elasticity pertains to income gain (reflecting ability enhancement) per dollar of public goods provided as income rises.  Here the counter-factual hypothetical is apparently a state of nature in which we have all the same people, with all the same innate talents, but none of the rudiments needed for a well-functioning economy.

Again, both models lead to a flat rate tax system with an assumed relevant elasticity of 1, based in each case on something that is unknowable.

Why would the relevant elasticities come out the same, causing the Smith benefit tax and the Mill equal sacrifice tax to look so similar?  This seems highly coincidental.  The main explanations that occur to me are:

1) Prominence and salience of 1 as a postulated elasticity.  Perhaps it feels safe and neutral to say that the posited trait grows at a steady rate with income, not faster or slower.

2) Prominence and salience of the flat tax result, which may look intuitively appealing (and also appear to have optical and political economy advantages), suggesting that the causal arrow ran in the other direction, i.e., from the intuitively favored tax rate to the mode of justification.

Either way, I draw an inference that we should not be all that confident that the equal sacrifice and benefit tax rationales are actually doing a whole lot of work.