Nonexclusive competition for a freelancer under adverse selection

A freelancer with a time constraint faces offers from multiple identical parties. The quality of the service provided by the freelancer can be high or low and is only known by the freelancer. The freelancer’s time cost is strictly increasing and convex. We show that a pure-strategy equilibrium exists if and only if the preferences of the high-type freelancer satisfy one of the following two distinct conditions: the high-type freelancer does not prefer providing his services for a price equal to the expected quality at the no-trade point; the high-type freelancer prefers providing his services for a price equal to the expected quality at any feasible trade point. If holds, then in equilibrium, the high-type freelancer does not trade, whereas the low-type may not trade, may trade efficiently, or may exhaust all of his capacity. Moreover, the buyers make zero profit from each of their traded contracts. If holds, then both types of the freelancer trade at the capacity in equilibrium. Furthermore, the buyers make zero expected profit with cross-subsidization. In any equilibrium, the aggregate equilibrium trades are unique. Our results extend to the case where the freelancer has more than two types if the buyers are restricted to offering concave tariffs.