The Perfect Haircut

Glen Whitman at Agoraphilia continues his efforts to use practical mathematics to unveil underlying truths about our day-to-day lives. This time, he argues based on economic reasoning that it is best to get a haircut that is a bit shorter than your ideal hair length. I am concerned, however, that his model leaves out an important fact.

I disagree with him about some of the specifics of the utility vs. hair length curve; in particular, I think that having hair that’s too short tends to be worse than hair that’s too long by an equal number of inches. Perhaps that is because my hair is just a bit longer than that dangerous “not quite a buzz cut” zone that Neal mentions in the comments, but it may be rather common because hair length is bounded below (by baldness) but not above. However, it is still a good idea to get a haircut that's a little too short for people whose disutility vs. hair length curves are asymmetrical in this way, as long as the disutility curve is continuous (as it should be). In fact, assuming that you get a haircut according to a regular schedule and you’ve decided on a certain length of time that you want between haircuts, you’ll minimize your total disutility by getting your hair cut so that your hair is just as bad right after the cut (on account of being too short) as it was just before the cut (on account of being too long). That’s the equilibrium point where the marginal benefits of cutting the hair longer (or shorter) equal the marginal costs.

The real flaw in Glen’s model reflects a very general and uncontroversial observation: people tend to notice things more when they change drastically than when they are hardly changing. Applying this piece of conventional wisdom to haircuts, we would suspect that people pay more attention to hair right after it has been cut. Indeed, this does seem to be true, at least in my experience. Only after getting a haircut am I likely to look at my hair in the mirror, think about how good it looks, and actually appreciate it. The rest of the time it’s just sort of there. My Total Utility is much more sensitive to my Hair Utility immediately following a haircut, so it makes sense for me to want my haircut to look good the day that I get it.

Hair is not just something that we consume privately for ourselves, though, it is something that we display to the world, in hopes of impressing others and attracting their admiration (or at least not doing the opposite). And other people also tend to notice your hair right after a haircut. When is the last time that someone has told you “I like your hair – has it been growing?” For people who see you regularly, at least, it’s an advantage to have good hair right after the cut, when it will catch their attention.

These one-time benefits of a good haircut must be included in a thorough analysis, in addition to the utility or disutility that accrues from having hair of a given length on any particular day. For most people, then, the ideal haircut is not as short as Glen (or the simple marginal costs = marginal benefits equation) would have you believe. Barbers who try to make your hair perfect as you walk out of their shop may actually be responding to their customers’ desires, rather than their ignorance.