The farther your hair length is from the “right” length, in either direction, the greater is your disutility from bad hair days. Too short and too long are both troublesome. Let’s suppose the problem is symmetrical, so that (for instance) hair one inch too long and hair one inch too short are equally undesirable. For simplicity, let’s say your hair grows one-half inch per week, and you get one unit of disutility for each inch of difference from the right length. If you’re on a 5-week haircut cycle, and you start off with the right length, your total disutility is:
0 + 0.5 + 1 + 1.5 + 2 = 5whereas if you got your hair cut an inch too short, your disutility would be:
1 + 0.5 + 0 + 0.5 + 1 = 3Clearly, you’re better off asking for the shorter haircut, since that minimizes your disutility.
But it’s unrealistic to assume every inch of difference corresponds to the same amount of disutility. I’d rather have two weeks of hair one-half inch too long than one week of hair a whole inch too long, because a whole inch is more than twice as annoying as a half inch. If your marginal disutility is proportional to (say) the square of inches, the comparison above becomes even more stark. Starting with the right length, you get disutility of:
0 + 0.25 + 1 + 2.25 + 4 = 7.5whereas if you started an inch too short, your disutility would be only:
1 + 0.25 + 0 + 0.25 + 1 = 2.5In such a case, having an optimal haircutting strategy matters a great deal more.
Now consider the hastened haircut effect. If there’s some threshold disutility above which you simply cannot stand it any longer, you’ll break from your planned cycle and get an early haircut. Say your threshold is 2 units of disutility, and you start with the right length. Then, using the numbers above, you’ll cut your cycle short and have a haircut after four weeks instead of five.
A cynical person might suggest that your barber or hairdresser, in sending you out of the shop with the “perfect do,” deliberately tries to short-circuit your cycle to squeeze more haircuts out of you. But I’m not a cynic on this one. I suspect hairdressers just don’t trust you to attribute your great hair two weeks from now to the great cut they gave you today.