Survey_KU_2005

Abstract

We want to characterize long-term consumption insurance contracts that competitive profit-maximizing principals offer to agents that cannot commit to honor these contracts. After the realization of income yt,i, but before consumption takes place, an agent is free to leave the principal and join a competitor. She takes the current income realization with her. We assume that moving is ”painful” to the agent, inflicting a disutility ?(yt,i) ? 0. For most of the paper, we will concentrate on the case ?(.) ? 0, that is, moving carries no direct cost. A principal has the ability to commit to long-term contracts with his agents, but has no ability to reach them in the future, once they have left for a competitor. In short, this is an environment with one-sided commitment.9 We now formulate a game of competition between principals, offering consumption contracts to potential movers and to agents already with the principal. We proceed directly to the recursive formulation of each individual principal’soptimization problem, and then to define a symmetric stationary recursive equilibrium. We thereby skip the step of first describing the game as unfolding sequentially. An agent enters the period with current state (y,w), describing her current income y and the expected discounted utility w from the contract she had been promised by the principal last period. The fact that utility promises w and the current shock y form a sufficient description of an agent’s state, in the sense that the resulting policy functions of the recursive problem induce consumption and investment sequences that solve the corresponding sequential optimization problem, has been demonstrated by Atkeson and Lucas (1992) for a private information economy and adapted to the environment presented here by Krueger (1999). Both papers borrow the idea of promised utility as a state variable from Abreu, Pierce and Stacchetti (1986) and Spear and Srivastava (1987). The objective of the principal is to maximize the contribution to his own lifetime utility (lifetime profit) from the contract with a particular agent. He is constrained to deliver the utility promise w by giving the agent current consumption c and utility promises from next period onwards, contingent on next period’s income realization, w0(y0). If the principal promises less utility from tomorrow onward in a particular income realization y0 than a competing principal, the agent will leave the location, and the principal makes zero profits from the contract with that particular agent from then on.10 We denote the utility promise by competing principals as U Out(y0), which the principal takes as given(but which is determined in equilibrium).