Assortative Matching, Reputation, and the Beatles Breakup

Consider Becker's (1973) classic static matching model, with output a stochastic function of unobserved types. Assume symmetric incomplete information about types, and thus commonly observed Bayesian posteriors. Matching is then assortative in these Bayesian 'reputations' if the expected output is supermodular (i.e. complementary) in the underlying types. We instead consider a standard dynamic version of this general equilibrium world, and discover a near total meltdown of Becker's global result. We show that as the production outcomes grow, assortative matching is neither efficient nor an equilibrium for high enough discount factors. The rough reason is that patient individuals (or a patient social planner) seeking to maximize their reputation can achieve a more informative spread by matching non-assortatively. Specifically, for a labour-theoretic link, we link the technology to where Becker's result breaks down. We show assortative matching fails around the highest reputation agents for 'low-skill concealing' technologies. Finally we turn from the failure in the large to a simple implication in the small. Namely, our theory implies the dynamic result that high-skill matches (such as the Beatles) eventually must break up for 'low-skill concealing' technologies. Our results owe especially to two findings: (a) convexity of the Bellman value function due to learning undermines static match supermodularity; and (b) for a fixed policy in an optimal learning exercise, the second derivative of the Bellman value function explodes geometrically at extremes. The property is used critically to decide a horse race between productive and informational efficiency as the discount factor rises to 1