Hierarchical Assignments: Stability and Fairness

We study a simple assigning workers to employers, where each pair of a worker and an employer has a potential joint productivity, and the complete information about the market is contained in the matrix of potential productivities. Under certain conditions that we specify, the market is hierarchical, in the sense that both workers and employers can be ordered according to ability, and the Pareto optimal assignment in terms of maximal total productivity is achieved by matching the top worker with the top employer and so on. In other words, we describe a market situation in which the above simple matching procedure is optimal. Some further properties of hierarchies are presented. We can state explicit values for the earnings in the worker optimal and employer optimal solutions. We further show that our hierarchy concept is descrete analogue to the Ricardian differential rent model of Sattinger (1979), and that the latter one can easily be derived from our model. We discuss the compatibility problems between fairness and stability of earnings and assignments. In particular, two notions of fairness that seem sensible in general fail to be stable in hierarchical markets. First, pairwise sharing the Pareto optimal product using a fixed sharing rule generates stable assignments only under a very restrictive assumption. Second, distributing the same amount to all workers preserves stability only in the extreme situation of an equal distribution of ability among workers. A uniform distribution of ability will always generate a smaller overall production than an uneaven distribution in Pareto optimum. we argue for another notion of fairness that turns out to be stable: an average between the worker optimal and employer otimal solutions. The model can be used to illustrate imperfect competition, economic growth and corruption.