, Volume 9, Issue 2, pp 271-289
Date: 03 Nov 2012

Utilities bounded below

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It is common to work with utilities which are not bounded below, but it seems hard to reconcile this with common sense; is the plight of a man who receives only one crumb of bread a day to eat really very much worse than the plight of a man who receives two? In this paper we study utilities which are bounded below, which necessitates novel modelling elements to prevent the question becoming trivial. What we propose is that an agent is subjected to random reviews of his finances. If he is reviewed and found to be bankrupt, then he is thrown into jail, and receives some large but finite negative value. In such a framework, we find optimal investment and consumption behaviour very different from the standard story. As the agent’s wealth goes negative, he gradually abandons hope of ever becoming honest again, and plunders as much as he can before being caught. Agents with very high wealth act like standard Merton investors.