Contracts versus Salaries in Matching

594 Workers and firms may bargain over general, multidimensional contracts; they may negotiate over health benefits, housing, retirement plans, etc. Substitutes, on the other hand, is the assumption commonly placed on firms’ preferences to guarantee the existence of stable matchings of workers and firms. In this paper I show that, when firms regard contracts as substitutes, bargaining over contracts can be embedded into a model of bargaining over wages. The economics of the embedding is straightforward, except for a small twist. When a firm and a worker negotiate over a contract, they may bargain over many dimensions. However, the Pareto frontier of contracts is, in a sense, “one dimensional”: what is better for the worker is worse for the firm. So Pareto optimal contracts may be viewed as salaries, with the better contracts for the firm meaning lower salaries, and the better contracts for the worker meaning higher salaries. The twist is that a firm’s ranking over contracts might be affected by the firm’s other hires. For example, health plan A may be better than B for a firm if it has many employees, but B beats A if it has few. When contracts are substitutes, it turns out that the ranking is not affected in this way. Hatfield and Milgrom (2005) present a model of two-sided worker-firm matching with contracts. A firm will hire a collection of workers and will negotiate a contract with each one of them. The model is a generalization of Kelso and Crawford (1982), where each firm and each worker negotiate a wage. Kelso and Crawford show that, when firms’ demands satisfy gross substitutes, the core of the matching market is nonempty. Hatfield and Milgrom show that, when the firms’ preferences over contracts satisfy their notion of substitutes, the core of the market is nonempty. I show that Hatfield and Milgrom’s model can be embedded into the model of Kelso and Crawford. Hatfield and Milgrom’s assumption of substitutability enables an embedding where firms’ demands for workers satisfy Kelso and Crawford’s notion of gross substitutes. As a result, the nonemptiness of the core follows from the argument in Kelso and Crawford, and their salary adjustment algorithm finds a stable matching of workers to firms, and a vector of supporting salaries. Hatfield and Milgrom’s paper is an elegant analysis of two-sided matching. It simplifies matching models and makes their relationship to auction models more transparent. Their paper contributes much more than showing the nonemptiness of the core when firms and workers can sign general contracts, and my result does not diminish