But something about that sounds fishy to you. “No thanks,” you say. “Could you please take the twenty percent off first, then apply the tax to the sale price? That way, I’ll pay less tax.”
Which way is a better deal for you? (Assume both are legal.) When confronted with a question like this, many people approach it additively. They work out the tax and the discount under both scenarios, and then do whatever additions or subtractions are necessary to find the final price. Doing things the clerk’s way, you reason, would cost you $4 in tax (8 percent of the sticker price of $50). That would bring your total to $54. Then applying the 20 percent discount to $54 gives you $10.80 back, so you’d end up paying $54 minus $10.80, which equals $43.20. Whereas under your scenario, the 20 percent discount would be applied first, saving you $10 off the $50 sticker price. Then the 8 percent tax on that reduced price of $40 would be $3.20, so you’d still end up paying $43.20. Amazing!
But it’s merely the commutative law in action. To see why, think multiplicatively, not additively. Applying an 8 percent tax followed by a 20 percent discount amounts to multiplying the sticker price by 1.08 and then multiplying that result by 0.80. Switching the order of tax and discount reverses the multiplication, but since 1.08 × 0.80 = 0.80 × 1.08, the final price comes out the same.
Considerations like these also arise in larger financial decisions. Is a Roth 401(k) better or worse than a traditional retirement plan? More generally, if you have a pile of money to invest and you have to pay taxes on it at some point, is it better to take the tax bite at the beginning of the investment period, or at the end?
Once again, the commutative law shows it’s a wash, all other things being equal (which, sadly, they often aren’t). If, for both scenarios, your money grows by the same factor and gets taxed at the same rate, it doesn’t matter whether you pay the taxes up front or at the end.
Please don’t mistake this mathematical remark for financial advice. Anyone facing these decisions in real life needs to be aware of many complications that muddy the waters: Do you expect to be in a higher or lower tax bracket when you retire? Will you max out your contribution limits? Do you think the government will change its policies about the tax-exempt status of withdrawals by the time you’re ready to take the money out? Leaving all this aside (and don’t get me wrong, it’s all important; I’m just trying to focus here on a simpler mathematical issue), my basic point is that the commutative law is relevant to the analysis of such decisions.
You can find heated debates about this on personal finance sites on the Internet. Even after the relevance of the commutative law has been pointed out, some bloggers don’t accept it. It’s that counterintuitive.
Maybe we’re wired to doubt the commutative law because in daily life, it usually matters what you do first. You can’t have your cake and eat it too. And when taking off your shoes and socks, you’ve got to get the sequencing right.
The physicist Murray Gell-Mann came to a similar realization one day when he was worrying about his future. As an undergraduate at Yale, he desperately wanted to stay in the Ivy League for graduate school. Unfortunately Princeton rejected his application. Harvard said yes but seemed to be dragging its feet about providing the financial support he needed. His best option, though he found it depressing, was MIT. In Gell-Mann’s eyes, MIT was a grubby technological institute, beneath his rarefied taste. Nevertheless, he accepted the offer. Years later he would explain that he had contemplated suicide at the time but decided against it once he realized that attending MIT and killing himself didn’t commute. He could always go to MIT and commit suicide later if he had to, but not the other way around.