Competitive adjustment processes in labor markets where firms and workers are heterogeneous but well informed are studied. A natural notion of equilibrium for such markets is defined, and a plausible adjustment process is shown under reasonable assumptions always to converge to an equilibrium; this allows a generalization of several existence results in the literature. Finally, the relationship between market institutions (such as who makes offers) and which of the range of equilibria that heterogeneity makes possible arises, is studied. Generalizing results of Gale and Shapley and Shapley and Shubik, it is shown that all agents on a given side of the market agree on which is the best equilibrium, and that the equilibrium that emerges is the one most favored by the agents on the side of the market that makes offers in the adjustment process. The process can also be viewed as an algorithm for transportation and optimal assignment problems.