Gentlemen Prefer Blands; or it all depends on the elasticities.

Simon Clark comments on the posting below by Santi Sanchez-Pages (Gentlemen Prefer Dumbs)

My colleague Santi sets out a very interesting analysis. As any good economist will tell you, it all depends on the elasticities. If v (value of match) = pq, and p depends on q, then write p = p(q) (with p' < 0) so v = p(q)q. Then dv/dq = qp' + p is positive if p'q/p, the elasticity of breeding probability with respect to female quality, which we label as e, is greater than -1 (recall that p' < 0).

If e > -1 (e.g. if p is constant) then the higher quality is not offset by the reduction in probability, so we would still have positive assortative matching (PAM). If e < -1, we get negative matching (NAM); high quality men will want to avoid high quality women as they are too unlikely to have children.

But e may be variable. Suppose q lies between 0 and 1 and p = 1 - q. Then v = q-qq (I can't do squares in Html!); simple calculus, or graphing v against q, shows that the highest value women have q = 0.5 and the worst 0 or 1. Women can be ranked by the absolute value of (q-0.5), so q = 0.25 is as good as q = 0.75. Then we would see NAM between the highest quality women and a representative half of the men (of all types) and PAM between the lowest quality women and the other half of the men. More like 'Gentlemen prefer blands'.

With equal numbers of men and women, whether some agents remain unmatched depends on whether they have a 'reservation quality' (as in 'I'm not that desperate!'). In the set-up above, if men will not accept v less than v*, single men will be low quality, and single women will have q outside the interval bounded by the two solutions to q - qq = v*. So we would observe spinsters who are either successful professional women too busy to breed or women ready to breed but too uneducated; an interesting area for empirical research.

As Santi says, it is true that we have a lot to learn from other disciplines, but the concept of elasticity can also be useful outside economics. Note the resemblance between v = pq ,and revenue = pq = price x quantity. If a man has a cost per unit of quality of c of providing 'satisfaction' to a high quality woman then v = (p - c)q, which can be thought of as total revenue less total cost. So maybe there are further parallels to be explored.