## Friday, October 22, 2004

### Optimal Sex Frequency

In the previous post, I discussed the diminishing marginal utility of sex. But in thinking further about the matter, it occurred to me that sex – and many other activities for that matter – may be subject to a somewhat different phenomenon. As the frequency of sex acts increases, it’s not just that each act is less satisfying than the previous one, it’s that the satisfaction from each and every act may decline. For instance, a person who has sex twice a week may appreciate each instance more (because of greater anticipation, built-up sexual tension, and so on) than a person who has sex every day.

For lack of a better term, I will refer to this phenomenon as “diminishing typical utility” (DTU). This is not exactly the same as diminishing marginal utility (DMU), at least DMU is often described. [See UPDATE below.] A person with DMU of sex might get utility of 12 from the first act, 9 from the second act, 7 from the third act, and so on. A person with DTU might get utility of 12 per act if he has sex once a week, 9 per act if he has sex twice a week, 7 per act if he has sex three times a week, and so on. Now, the same person might be subject to both DMU and DTU, but I suspect that DTU is the more important effect for a person having sex on a regular basis. For such a person, there is not a “reset” point – say, Sunday evening – at which marginal utility jumps back up to its initial high value. Instead, what matters is the frequency of sex in the cycle. A person who has sex twice a week will typically have waited about 3.5 days between acts, whereas a person who has sex seven times a week will have waited only about one day.

With DMU, having more frequent sex always increases your satisfaction (so long as the marginal utility is still positive). But with DTU, having more frequent sex creates offsetting effects: on the one hand, you enjoy the additional act, but on the other hand, each act has lower significance. So, if you’re a person with DTU of sex, how much sex should you have?

Calculus to the rescue! Let x equal your number of sex acts per time cycle (say, a week). Let s(x) equal your satisfaction per act of sex. DTU implies that s(x) has a negative first derivative, s'(x) < 0. Your total satisfaction from sex is xs(x). To maximize your satisfaction, find the first derivative (using the product rule) and set it equal to zero:
xs'(x) + s(x) = 0
or, rearranging,
s(x) = – xs'(x)
This optimal-sex-frequency condition has a nice interpretation. The left-hand side is the satisfaction you get from one act of sex considered in isolation. The right-hand side is the loss in satisfaction per sex act that results from increasing the frequency ( – s'(x)), multiplied by the number of sex acts whose satisfaction is reduced in that manner. So the condition says to increase your frequency of sex until the added satisfaction from having sex once more is exactly counterbalanced by the diminished satisfaction of all sex acts together.

(Aside for econ majors: This problem is mathematically equivalent to revenue maximization. To maximize revenue, expand output until the revenue received from your last unit – that is, the price – is just offset by the lost revenue from having lowered your per-unit price on all units in order to sell more units. Note, however, that revenue is not profit.)

However, the optimality equation above will only have a solution if greater frequency diminishes your satisfaction from each act, i.e., if s'(x) < 0. If you, like Steve, have s'(x) > 0, then the optimal response is what economists call a corner solution: have all the sex you possibly can.

Also, if sexual partners have different solutions to the maximization problem, then the stage may be set for a relationship cycle.

[UPDATE: In pondering a bit more, I've decided that DTU is actually a special case of DMU, but it doesn't fit with the usual description of DMU. Technically, marginal utility is defined as the change in total utility that results from an increase in quantity. With that definition, DTU is indeed a form of DMU, because the reduction in satisfaction per act is part of the change that results from an increase in quantity. But DMU is often described loosely as the satisfaction attributable to the very last unit of quantity -- e.g., the satisfaction you get from eating your third slice of pizza. If DMU is construed in this narrower way, then DTU appears to be a distinct phenomenon.]

Steven Horwitz said...

Heh. I love it. Now I'm a sexual corner solution. I think I've always known that about myself, now I have Science on my side. ;)

David said...

Of course, Glen is assuming ceterus paribus--every sex act, from the level of arousal to the partner to the lighting to the position to the amount of time spent, is exactly the same. That makes sense because it most certainly gets less interesting. Even if you change positions during a single act or change locations or change partners, the marginal utility is probably decreasing (though decreasing at a lesser rate).

But we don't live in that world and I doubt that Steve is having ceterus paribus sex. Each act of sex, therefore, should be considered a different good, unless you're one of the boring types that never change your sexual habits, in which case, there are stores for you.

Most people, I think, face the situacion that Steve faces: increasing marginal utility. But Glen's solution that we (and I count myself among these people) should have as much sex as we can doesn't hold water. First, because we need a partner, we face genuine uncertainty. We can have bad sex (and yes, there is such a thing as that, even for guys). Or maybe it's just not as good as we'd like. And that doesn't include the costs--condoms, STD risks, pregancy risks, time and energy, etc. The last two are of particular importance; between the act itself, clean up, (hopefully) foreplay, listening to her talk after its over (kidding, I love that part) there's a lot else that could be done. And there's still risk of heart attack.

What we CAN do is rank sex. We get little by ranking each hour of sex because we have to go through the first hour to get to the second, but we rank sexual states all the time. We prefer this position over that one, this person over that one, this number of people over that one, this surface over that one and so one.

Getting all this information requires research. Because I'm assuming that people face increasing marginal utility and they have reasonably active imaginations about this (and we are good at guessing what we like), the more research you do, the better your ranking is. If you have sex in position A your whole life, then you can't include a ranking of your taste of position B. However, people who try lots of things will discover some mind blowing combination and put that on the top of their rank. So the poll I want to see answers which group has the longer list of rankings (and thus the more varied, and theoretically more satisfying, sex life).

Glen Whitman said...

I don't think I need to assume perfectly identical acts of sex, any more than the standard consumption model needs to assume perfectly identical apples, slices of pizza, etc. There is some tolerance for difference even in a class treated as homogeneous for analytical purposes. But more importantly, among goods that are relatively close substitutes, having some of one good can diminish the marginal utility of a unit of another good. For instance, if I already have some butter, that reduces the added value I get from a unit of margarine. So even if sex acts may differ substantially, DMU or something like it will still hold as long as the sex acts are close enough substitutes for each other -- e.g., this position is a reasonable substitute for that position.

It's true, of course, that costs I haven't mentioned (like condoms, clean-up, etc.) would tend to reduce the optimal frequency of sex, even for the person with increasing marginal utility. But what my analysis shows is that even if sex had none of these costs, it would still be optimal for a person with DMU to limit the frequency of sex.

Neal said...

Heh. Cute choice of variables. Reminds me of the front of a catalog aimed at college guys, with a comely young blonde holding some serious-looking textbooks and using a chalkboard pointer to point out a calculus equation: S.e^x = f(u^n). "Hey, wait!" I was thinking. "Where's the dx after S.e^x? Oh, *I* get it..."

Anonymous said...

My man has diabetes, so I never get satisfied.

David Friedman said...

"But DMU is often described loosely as the satisfaction attributable to the very last unit of quantity -- e.g., the satisfaction you get from eating your third slice of pizza."

Loosely and inaccurately. Your point about sex is the normal case of diminishing marginal utility, not something special.

Consider oranges. The marginal utility isn't the pleasure from eating an eighth orange this week. It's the difference between the pleasure of eating eight oranges a week and the pleasure of eating seven. If you think in terms of rates, it should be obvious that there is nothing special about the eighth orange--no reason why it should give you less utility than the seventh. But there is a reason why you get less pleasure per orange the more oranges a week you are eating.

Anonymous said...

Key assumption sex formula is in base 2. On base 3 derivative your conclusion is reverse

Anonymous said...

The economy of sex? Clever, but sad. Here DMU is not the measure of satisfaction but just the expression of a total lack of understanding what sex really is.

Reticuli said...

Micro-economics is soooooooo hot.

I believe the average optimal frequency that's been arrived at during clinical studies of the whole (mostly sexually starved American population) is like 1.5 time per week. But that is highly dependent on the individual and relates to serotonin levels versus dopamine levels trading off in various quantities and at varying speeds. A "crash" is likely an indication you're having sex too frequently or you have an underlying chemical imbalance that requires SSRIs to compensate for. If no sex for at least a week or (better yet) a month produces the same effect without SSRIs, then you probably need anti-depressants. Otherwise, sex frequency can be adjusted to compensate as needed. Like econ productivity and labor hiring for a firm, you really won't know until you've gone over. So if post-coital mood is still groov'n, you can try slightly more frequently if that’s possible. Other medications, exercise, and a variety of other health/wellness factors all seem to contribute. So this can be variable for the same person depending on time of day, season of the year, amount of (other) physical activity, diet, and on and on. If you want to dump/divorce someone right after sex when they weren't even "bad" (bad bad not baaaaad), then take a few days off from each other and see if you start the long'n again.