This paper develops and extends a dynamic, discrete time, job to worker matching model in which jobs are heterogeneous in equilibrium. The key assumptions of this economic environment are (i) matching is directed and (ii) coordination frictions lead to heterogeneous local labor markets. We de- rive a number of new theoretical results, which are essential for the empirical application of this type of model to matched employer-employee microdata. First, we o¤er a robust equilibrium concept in which there is a continu- ous dispersion of job productivities and wages. Second, we show that our model can be readily solved with continuous exogenous worker heterogene- ity, where high type workers (high outside options and productivity) earn higher wages in high type jobs and are hired at least as frequently to the better job types as low type workers (low outside options and productivity). Third, we demonstrate that the tractability of this framework is enhanced by analyzing and proving the equivalence of "seller auctions" and "buyer posting". We also prove a related result concerning the equivalence of buyer posting and seller posting when buyers di¤er continuously. Finally, we show that all of these results preserve the essential tractability of the baseline model with aggregate shocks. Therefore, we o¤er a parsimonious, general equilibrium framework in which to study the process by which the contin- uous dispersion of wages and productivities varies over the business cycle for a large population of workers with continuous dispersion of unobserved worker types.