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.


I appreciate your point on the way the mind functions - but as you mention in your final paragraph, I don't think we should be looking for ways to make paying higher taxes more confusing to the public.


Why not 9.9%?
Looking prices in shops, this would be definitely efficient.


Or just omit the sales tax rate and amount from the sales receipts in the first place, and mandate the tax burden be incorporated into the selling price.


Now that I think about it, what would be interesting is continuously compounded interest at a rate of e%. That would give us future values of V = initial value * e^(e*time) ... and what would happen if you taxed your gains at a rate of e^2?


not a very democratic post: confuse the masses so the vanguard can more effectively lead- the more 'rational' approach would be to have better civic education so that citizens would learn the maxim: everybody pays and everyone gets use

Mike B

This is one of the reasons I REFUSE to get a EZ(Government Surveillance)-Pass or set up automatic bill payment. I want to be fully aware every time I am assessed a charge or fee so that I will be spurred to continually re-evaluate my economic choices. It is also the reason I tend to avoid smart card based transit cards that just need to be "tapped".

This is also why I prefer to the common use of dollar bills over dollar coins. Paying with bills has higher salience than paying with coins. When I traveled to France about 15 years ago they had a handy 10 Franc coin and conveniently enough the price in most vending machines for a can of soda was 10 Francs. After a few days of casually feeding the machines I came to the realization that I was constantly paying $2 for 50 cent cans of soda!! I mean I was aware of the exchange rate, but plunking in one messily coin didn't register as strongly had I been forced to feed in bank notes or multiple smaller denomination coins.



Shouldn't this be a criticism, not a defense? If less salience is what you want, we shouldn't have sales taxes at all. We shouldn't even tax individuals. We should tax employers. We can have the IRS do all the math so that taxpayers wouldn't be bothered with calculating their taxes. We can hide the formulas in vaults to which only the IRS has access.


People fear what they don't understand.

Most people can't even spell, much less figure out e^2.

Artie Gold

Dazzle 'em with "bushwa" (avoiding the inelegant indelicacy).

Of course, not only do you get the value of e wrong in the text, you also understate the whole concept of the nature of the real numbers...

Innumeracy for innumerates?


There's a big hole in the logic around EZ-Pass. More people use the toll roads with EZ-Pass because there is less of a time delay for toll collection. Dollars are not the only cost of a toll.


Also the myriad of federal, state, and local taxes on income and expenditures for consumers and businesses hides the real tax rates.

What if we had just one rational tax?

Imad Qureshi

I use I-Pass in Chicago because I have to pay double if I pay cash.


Isn't the example from the article (VAT) much better than your e^2 idea?

Matt J

This idea depends on the motives for taxation. If the motive is purely to rasie government revenue, then yes we want to be sure the tax doesn't distort behavior too much and create too much deadweight loss. Income taxes come to mind here.

However, if the purpose of the tax IS to affect behavior, then I think that ensuring the salience is the most important part! If we ever get around to taxing carbon, I hope it is a shiny round number. Cigarette taxes raise money, but they also are (presumably) in place to deter people from smoking. Even sales tax - one could argue that we should be championing the salience of sales tax to encourage savings in lieu of consumption.


The same is true for using electronic money (credit card, debit card, paypal, etc) , in general. People tend to spend more when such an option is available. I can see the positives easily, convenience, security, etc. But do the benefits outweigh the costs such as overspending? I should look up articles and papers on this subject.

Now should the govt. confuse the citizenry, just to take advantage of their poor calculation skills? This is getting close to an ethical issue.


Ian Kemmish

Maybe you have too much money, Mr Ayres. If I'm somewhere where the exchange rate isn't something simple, I either:

a) round up, or

b) make no impulse purchases. Through fear, not through rational caution.

VAT would appear to be a nicely non-salient tax for you to get your teeth into. Most people over here can't remember whether it's applied to biscuits and not cakes, or to cakes and not biscuits. (Or whether the European courts decided that for tax purposes a Jaffa cake was actually a cake or a biscuit.)


Of course, when it comes to computers, there is no such thing as an irrational number. The amount of precision you get is proportional to the number of bits used to represent the number. Presumably, supporting more precision in the tax rate would require a re-write of the IRS's software, to say nothing of the problem of deciding how much precision would actually be needed to support future inflation, etc. Any good feelings gained by obscuring the tax rate would likely be wasted on the subsequent boondoggle of actually implementing it.


I've never seen a drop of #2 fuel oil, but I can assure Steven Levitt that paying over $4 a gallon, nearly twice what it had been the previous season, enraged me as I wrote the checks and every time the furnace kicked on two winters ago. And that was before I learned that it was all due to market manipulation.

I believe the thinking that characterizes the appeal of nontransparency in taxing predates the Great Fiscal Awakening of 2008-2009. Many of us are scrutinizing prices and costs--taxes, surcharges, gotchas of every stripe--in a way we haven't done in decades, and we're asking questions and challenging rates and fees. Trying to slip us the mick by coding taxes in a non-numerical format doesn't seem a very smart way to make dunning the public more palatable in this environment. It is, however, a great way to earn forensic examination of said taxes and thinking by Rachel Maddow, and a scabrous chaser of condemnation by Keith Olbermann.



I agree that the change from a round number makes it less likely to be thought about. When Massachusetts was at 5%, I always knew exactly how much tax I owed. Now that it's 6.25%, I don't bother thinking about it.

Eric M. Jones

Irrational numbers are under-used in our world. After all "e" is the basis of natural growth curves.

I once tried to decide when I should sell my computer and buy a newer faster more wizzbang one. Quantifying how "fast" or "powerful" a computer is is not easy, but I figured that for a given amount of money ($1000 was my limit), if I could buy a computer that was "e" times more powerful (by whatever measurement), I'd be following a natural growth curve that would satisfy me.

So far it works well.