On the Rank, Generalized Lorenz and Overtaking Criteria for Evaluating Stochastic Income Regimes

Abstract

This essay is concerned with the evaluation of income regimes, that is a set of income classes and the stochastic process which governs the individual's position among the income classes. We follow Champernowne [4], Theil [21], Shorrocks [18;19], Bartholomew [2], Geweke, Marshall, and Zarkin [9], and Conlisk [5] in characterizing mobility as a Markov chain. Here, states of a Markov chain correspond to positions in the income classes. Utilizing the ergodic theorem for Markov chains and propositions from the theory of decisionmaking under uncertainty, we derive necessary and sufficient conditions on the income classes for one such income regime to be preferred to another in terms of the criterion. Specifically, we show that the rank dominance condition (Saposnik [17]) is necessary and sufficient for there to be a critical time at which occurs for each agent. This time depends on the agent's von Neumann-Morgenstern utility function which is merely required to be monotone increasing in income. Moreover, versions of the generalized Lorenz criterion (Shorrocks [20]) provide necessary and sufficient conditions for there to be a time at which uniform overtaking occurs, this time being the same for all agents provided the agents are risk averse. Among the applications of this model to individual choice, one may consider a new college graduate deciding among employment opportunities. In addition to starting salaries, an applicant should consider the firms' entire salary schedules as well as prospects for promotion within each firm. Alternatively, one may consider a musician who must choose between a back section position with a leading orchestra or a more prominent position with a less highly regarded orchestra. Again, one may be wise to consider the entire compensation schedule as well as the prospects for