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This paper studies matching in vertical networks, generalizing the theory of matching in two-sided markets. It gives sufficient conditions for the existence of stable networks and presents an algorithm for finding two of them. One is the best stable network for the agents on the “upstream” end of an industry. The other is best for the agents on the “downstream” end. The paper describes several properties of the set of stable networks and discusses applications of the theory to the design of matching markets with more than two types of agents and to the empirical analysis of supply chains. (JEL C78, D85, L14) The woollen coat, for example, which covers the day-labourer, as coarse and rough as it may appear, is the produce of the joint labour of a great multitude of workmen. The shepherd, the sorter of the wool, the wool-comber or carder, the dyer, the scribbler, the spinner, the weaver, the fuller, the dresser, with many others, must all join their different arts in order to complete even this homely production. —Adam Smith (1776)
Michael Ostrovsky (Thu,) studied this question.