In a recent post, I discussed Gianni De Fraja’s paper in which he proposed that sexual selection shaped the nature of conspicuous consumption by men. In his model, conspicuous consumption by men serves as a fitness indicator to women. Low and high quality men signal their differing wealth “honestly” (under certain conditions) as the consumption level of the high quality men is too large a handicap for the low quality men to copy.
One of the unsatisfying elements of the paper (although through no fault of De Fraja’s) was that the conditions under which high and low quality men signal honestly were not readily interpretable. The mathematics were too ugly.
De Fraja based his model on two models developed by Grafen (here and here), which were in turn the first mathematical demonstration that Zahavi’s handicap principle was theoretically sound and could work as a stand-alone process. Grafen’s first model was a simple game theoretic model for which he found the equilibrium. The second model was a population genetic model that built on the first by being explicit on how women used the information they obtained about the males' quality. It also, as the name suggests, incorporated a genetic basis. Despite the differences in the two models, Grafen considered that the results from the population genetic model supported the use of the simpler game theoretic model and that the extra complications in the population genetic model did not negate the results of the other.
One virtue of Grafen’s game theoretic model is that it is possible to interpret the conditions under which it works. So, instead of trying to pull the conditions from the complicated mathematics of De Fraja’s model (for the moment), the conditions of Grafen’s game theoretic model are worth a look.
Putting Grafen’s model into human terms, it had three elements. First, men vary in quality (say, wealth), which women cannot observe. If they could see it, women would use it as a basis for their choice. Second, men vary in their level of conspicuous consumption, which is a function of their quality. Third, women infer the man’s quality from the level of conspicuous consumption and use it to decide their choice of mate. The fitness of a male depends on his true quality, his level of advertising (which is costly) and the woman’s perception of his quality. A woman’s fitness will depend simply on how accurate she is in inferring true quality.
Grafen showed that in this model an equilibrium exists where higher quality men advertise more than low quality men and the women use this information to correctly infer their quality. The condition for this equilibrium is that the marginal cost of advertising should be higher for worse males.
The question becomes whether this condition could exist in the conspicuous consumption example? On the one hand, a BMW costs the same to anyone who buys it. The marginal cost appears the same to both low and high quality men. But suppose that the rich man can buy the BMW with cash. The poor man needs to take out a loan, max out his credit card and hock his watch. He will be paying high interest on the loan and credit card and will need to pay extra to get his watch back. As a result, even though they are both buying the same advertising, the BMW, the marginal cost of that advertising, is higher for the poor man. This condition for the handicap could hold. The condition also holds where the poor man cannot afford the BMW no matter what he does. His marginal cost at that point is effectively infinite. In equilibrium, the rich guy will pick a level of advertising that will simply be too much for the poor chap to match.
This condition is important. Previous mathematical attempts to explain the handicap principle had generally not succeeded as they had not incorporated this higher cost of the handicap for lower quality males. In this interview, John Maynard Smith explains how his earlier work on the handicap principle had failed to support Zahavi’s claims for this reason.
There is actually an economics model which mirrors this situation (and pre-dates Grafen’s model by 17 years) - the job market signalling model of Michael Spence. The model works similarly. Suppose there are low and high quality employees and they need to signal their unobservable quality to an employer. Spence showed that there could be a separating equilibrium where each signals their quality accurately through their level of education. The condition for this is that education must be more costly for low quality people. This makes sense - it is easier to learn something and pass the tests if the person is higher quality.