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Utilities bounded below

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Abstract

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.

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Notes

  1. ... if the money was owed to legitimate lenders ...

  2. ... as in the case of disgraced SocGen trader Jerome Kerviel..

  3. As Kenneth Arrow has argued, it is absurd to imagine that a utility should be unbounded above; for then you would prefer an infinitesimal chance of gaining more wealth than exists in the universe to the certainty of gaining all the wealth you could consume in ten lifetimes. We may accept unbounded utilities if they lead to tractable analyses, but should be wary if our conclusions are critically dependent on the exact nature of the infinite asymptotics.

  4. If the agent was reviewed while \(w \ge 0\), he would be allowed to continue, so this review would have no effect.

  5. The process \(v(w_{t \wedge \tau })\) has to be a supermartingale for any stopping time \(\tau \), and a martingale for the optimal \(\tau \). If there was an upward jump of \(v^{\prime }\) at some point, then the supermartingale property would not hold because there would be an increasing local time contribution in the Itô expansion of \(v(w_t)\).

  6. In this, our problem differs from the situation considered in Karatzas et al. (1986), where bankruptcy of the agent is an allowed eventuality, but negative wealth is not. In the study of Karatzas et al. (1986), bankruptcy occurs at the (previsible) first time that wealth hits zero.

  7. In more detail, we know from our study of the dual value function \(J\) that for small arguments \(J\) is close to the dual value function of the standard Merton problem; accordingly, the value function for high wealth levels will be close to the Merton value function. We therefore impose the condition \((1-R)h(y^*) -y^* h^{\prime }(y^*) = 0\) at the upper boundary point \(y^*\).

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Acknowledgments

Financial support from the Cambridge Endowment for Research in Finance is gratefully acknowledged. Helpful comments from our discussant, Albert Menkveld, and other participants at the Cambridge Finance-UPenn-Tinbergen meeting June 2012 have also been most welcome.

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Correspondence to L. C. G. Rogers.

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Muraviev, R., Rogers, L.C.G. Utilities bounded below. Ann Finance 9, 271–289 (2013). https://doi.org/10.1007/s10436-012-0212-3

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