Time and Again

In her latest column, Marilyn vos Savant answers a reader's question about overtime pay options. "Our employer gives us a choice of overtime pay: 1) pay at 1½ times our hourly rate; or 2) compensatory time off at 1½ times our overtime hours worked [I assume this is paid vacation time]. My co-workers say comp time is a better deal, because no taxes dilute the time off: One gets 100% of what one earns. I argue that comp time is taxed too. Which choice of overtime (personal preference aside) has more 'bang for the buck'?"

Marilyn responds: "Getting overtime pay is a way better deal. The tax argument is weak because that logic justifies not even working at all. You could say, 'Why should I trade my hours for pay if I must give a part of that pay to the government when I could just stay at home and keep all of my time for myself?' In short, saving on your taxes is a financially unsound reason not to work, regardless of the number of hours."

This kind of reductio ad absurdum argument doesn't work very well in situations where people's preferences can change at the margin. What's true for the last hour of work is not necessarily true for the first. It's entirely possible that a tax rate that doesn't deter you from working your first hour will deter you from working your 40th hour. Nonetheless, it's true that the tax issue isn't really important in this case - although the reader, not Marilyn, gives the right reason: the tax is assessed on vacation pay, too. But Marilyn's claim that taking the overtime pay is a "way better deal" cannot be justified.

For simplicity, let's suppose that vacation days are taken the same week that an overtime hour is worked. If you take option A (overtime pay), you work 41 hours (40 regular hours and 1 overtime hour). If your wage is X, then your pay is 40X + 1.5x = 41.5X. The average wage is 41.5X/41 = 1.0123X. If you take option B (extra vacation days), you work 39.5 hours (40 regular hours, minus 1.5 vacation hours, plus 1 overtime hour). But you get paid for 40 regular hours (the overtime pay has been sacrificed for the extra vacation time), so your pay is 40X. The average wage is 40X/39.5 = 1.0127X.

Notice that the average wage is *higher* if you take the extra vacation time. Not much higher, granted -- if your wage is $10/hour, we're talking about less than half a cent. But that, indeed, is the point: there is almost no difference between the two policies, in terms of the average wage. And what negligible difference there is in the average wage points toward the extra vacation time. On what grounds could Marilyn claim that overtime pay is a "way better deal"? (Note that I'm not claiming that taking the overtime pay is a worse deal. It does result in a lower average wage, but total compensation is higher because of the larger number of hours worked. My point is that this is not clearly better or worse.)

Allow me to run a couple of objections off at the pass. First, maybe I'm incorrect in thinking the "comp hours" are paid vacation hours. Maybe they're days you can take off without pay. In that case, I agree that option B isn't very good -- it lets you work an extra hour now (with no pay) in return for a reduction in your work hours later (also with no pay). In fact, that offer seems so lousy that I can't imagine that's what the reader meant. If someone has another interpretation, I'd love to hear it.

Second, it might be objected that I should look at the marginal difference between the two options rather than looking at the average wage. Okay: The difference in hours between the options is 1.5 hours, and the difference in pay is 1.5X. So the marginal wage is 1.5X/1.5 = X, the regular hourly wage. The question, then, is whether you want to sell a little bit more labor for the same wage you've been selling it for all along. Is that obviously a good (or bad) deal, as Marilyn suggests? I think not. It looks like Marilyn has fallen for the fallacy of argumentum ad logicam: because the tax argument was defective, she thought the choice it was advanced to defend was also defective.

(BTW, Marilyn's column appears online here, but for some reason the letter in question does not appear. I read it in Parade magazine.)