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A Defense of Irrational Taxation?

Here’s a behavioral puzzler: Why might it be more efficient for Connecticut to change its sales tax rate from 6 percent to e^2 percent ?
Or more generally, why might using irrational numbers as tax rates be less distortionary than rational tax rates?
A hint comes from a great article by Amy Finkelstein, “E-ZTax: Tax Salience and Tax Rates.” Her simple and powerful idea is that as the salience of tax rates declines, taxes will produce fewer distortions because taxpayers will not pay as much attention to the taxes.
The E-ZPass system is a perfect context for her to examine this hypothesis because E-ZPass users (she finds) pay less attention to tolls than people who have to pony up the cash from their wallets or purses. Comparing E-ZPass highways to non-E-ZPass highways, she finds that as the proportion of drivers making electronic payments increases “toll rates are 20 percent to 40 percent higher than they would have been under manual toll collection.”
High salience prices can drive us crazy. Levitt has written that one reason the public was so upset about high gas prices was that they have to spend so much time standing at the pump and watching the higher price. High pump prices are the antithesis of EZ-Pass pricing.
The “out of sight, out of mind” effect suggests that policies to lower salience tax might reduce consumption distortions. I find it liberating to buy goods in foreign currency when I have difficulty converting the price into dollars. So to begin with, sales tax rates that are nice round numbers, like 10 percent, are likely to be more distortionary (than rates with many decimals) because it is so easy calculate the tax burden.
Taking this logic a step further leads to the perverse idea of using irrational numbers for tax rates. Since few Americans know that Euler’s number (e) is approximately 2.718, stating the sales tax rate in terms of e just might be lower salience. Classical economics would suggest that a tax rate of e^2 percent (approximately 7.39 percent) would produce higher distortions than a tax rate of 6percent because generally the higher the rate, the higher the dead weight loss. Finkelstein’s E-ZTax article makes me think that higher but less salient rates might be an exception to this rule.
By the way, there is no shortage of irrational numbers; there are an infinite number of irrational numbers between any two rational numbers.
Of course, as matter of political economy, we might as a society want to keep our taxes highly salient (even if it increases the dead-weight loss of taxes) to make sure that our representatives feel more constrained when deciding whether or not to hike our rates 20 to 40 percent.