We use the theory of abstract convexity to study adverse-selection principal-agent problems and two-sided matching problems, departing from much of the literature by not requiring quasilinear utility. We formulate and characterize a basic underlying implementation duality. We show how this duality can be used to obtain a sharpening of the taxation principle, to obtain a general existence result for solutions to the principal-agent problem, to show that (just as in the quasilinear case) all increasing decision functions are implementable under a single crossing condition, and to obtain an existence result for stable outcomes featuring positive assortative matching in a matching model.
We study markets in which agents ﬁrst make investments and are then matched into potentially productive partnerships. Equilibrium investments and the equilibrium matching will be eﬀicient if agents can simultaneously negotiate investments and matches, but we focus on markets in which agents must ﬁrst sink their investments before matching. Additional equilibria may arise in this sunk-investment setting, even though our matching market is competitive. These equilibria exhibit ineﬀiciencies that we can interpret as coordination failures. All allocations satisfying a constrained eﬀiciency property are equilibria, and the converse holds if preferences satisfy a separability condition. We identify suﬀicient conditions (most notably, quasiconcave utilities) for the investments of matched agents to satisfy an exchange eﬀiciency property as well as suﬀicient conditions (most notably, a single crossing property) for agents to be matched positive assortatively, with these conditions then forming the core of suﬀicient conditions for the eﬀiciency of equilibrium allocations.