Frictional assignment, Part II: Infinite horizon and inequality

In an infinite-horizon economy with matching frictions, I study the efficient assignment between workers of different skill levels and machines of different quality levels. Under some restrictions I show that the efficient allocation assigns a unique machine quality and market tightness to each skill, and that the assignment is saddle-path stable. The efficient assignment is not necessarily positively assortative and efficient wages do not necessarily increase with the skill level. Nevertheless, the social value of workers always increases with the skill level. I then show that the efficient allocation can be decentralized by a market mechanism, in which the firms direct workers' search by announcing and committing to the machine quality, the skill level they intend to hire for such machines, and the time-path of wages. Finally, I calibrate the model to the US data and examine how a skill-biased technological progress affects the assignment and inequality.