Consistency of the doctor-optimal equilibrium price vector in job-matching markets

In a many-to-many variation of Kelso and Crawford's (1982) and Kojima et al.'s (2019) job-matching model, we prove that the doctor-optimal equilibrium price vector satisfies a striking property termed consistency. It states that the price vector remains an equilibrium price vector in the market where an arbitrary doctor reduces her capacity by one. Combining consistency with Kojima et al.'s (2019) result, we delineate how the set of equilibrium price vectors moves when a doctor reduces her capacity by one: the set moves up but still contains the original doctor-optimal equilibrium price vector. We apply this result to establish comparative statics of welfare at equilibrium. As a byproduct, we also provide a simple and short proof of the claim that the doctor-optimal stable mechanism is a pivotal mechanism under quasi-linear preferences.