​Two-sided productivity heterogeneity, firm boundaries, and assortative matching

We consider a market where each firm is created by the combination of two complementary assets that are heterogeneous in their productivity. After assets match endogenously, their owners choose between two ownership structures: centralized organization (integration) and arm's length organization (nonintegration). Our main focus is on the interplay between productivity heterogeneity and firm boundary decisions. When firms choose between distinct ownership structures, the standard single-crossing condition that guarantees positive assortative matching may fail to hold. We provide a novel condition—the congruent marginal contributions property—which guarantees monotone matching with respect to asset productivity. Furthermore, we provide conditions under which integration at the bottom of the productivity ladder is the market equilibrium; an organizational pattern that has been largely unexplored by the theoretical and empirical literature. We investigate the effect of model primitives on the equilibrium distribution of output. Moreover, our model offers interesting testable implications regarding firm boundary decisions.