Equilibrium in Labor Markets with Few Firms

TL;DR

This paper analyzes equilibrium existence in labor markets with few firms using combinatorial models, focusing on additive and social network-based production functions, and explores conditions for stable competition outcomes.

Contribution

It introduces two classes of production functions for firms, analyzes equilibrium existence conditions, and extends findings through empirical studies for multiple firms.

Findings

Pure subgame perfect equilibrium exists with two firms under certain conditions.

Equilibrium may not exist with more than two firms, but empirical analysis provides insights.

Stability can sometimes extend to coalitional stability, affecting profit distribution.

Abstract

We study competition between firms in labor markets, following a combinatorial model suggested by Kelso and Crawford [1982]. In this model, each firm is trying to recruit workers by offering a higher salary than its competitors, and its production function defines the utility generated from any actual set of recruited workers. We define two natural classes of production functions for firms, where the first one is based on additive capacities (weights), and the second on the influence of workers in a social network. We then analyze the existence of pure subgame perfect equilibrium (PSPE) in the labor market and its properties. While neither class holds the gross substitutes condition, we show that in both classes the existence of PSPE is guaranteed under certain restrictions, and in particular when there are only two competing firms. As a corollary, there exists a Walrasian equilibrium in a corresponding combinatorial auction, where bidders' valuation functions belong to these classes. While a PSPE may not exist when there are more than two firms, we perform an empirical study of equilibrium outcomes for the case of weight-based games with three firms, which extend our analytical results. We then show that stability can in some cases be extended to coalitional stability, and study the distribution of profit between firms and their workers in weight-based games.