Abstract
Lotteries with infinite expected utility are inconsistent with the axioms of expected utility theory. To rule them out, either the set of permissible lotteries must be restricted (to exclude, at a minimum, “fat-tailed” distributions such as that underlying the St. Petersburg Paradox and power laws that are popular in models of climate change), or the utility function must be bounded. This note explores the second approach and proposes a number of tractable specifications leading to utility functions that are bounded both from above and below. This property is intimately related to that of increasing relative risk aversion as first hypothesized by Arrow (1965).
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Notes
Following Cramer’s (1728) description of what has become known as the “St. Petersburg Paradox,” if an agent receives a payoff of \(2^{n-1}\) with probability \(1/2^{n}\), \(n=\{{1,2,\ldots }\},\) the mathematical expectation of this lottery is infinite, though, as Cramer puts it, “no reasonable man would be willing to pay 20 ducats as equivalent.” This discrepancy is not so much a logical paradox than a challenge to the notion that expected value is an accurate guide to empirical decision making (Menger 1934).
Cramer also considers a linear utility function that is somewhat arbitrarily truncated at \(\overline{x}=2^{24}\).
Menger (1934) discusses the use of a bounded utility function but does not consider it a satisfactory resolution and instead conjectures that agents tend to underestimate small probabilities.
In particular, suppose \({\mathbb {E}}[u( X )]=\infty \) and \({\mathbb {E}}[ {u( Y )} ]<{\mathbb {E}}[ {u( {Z} )} ]<\infty .\) Then, \(\alpha X + ( {1-\alpha } )Y\succ Z\) for every \(\alpha \, \in \, (0,1),\) in direct violation of von Neumann and Morgenstern’s axiom (3:B:c).
In fact, this insight comes in form of a footnote in the second edition of his book. In the first edition, he argues that an assumption of bounded utility, while reasonable, “would entail a certain mathematical awkwardness in many practical contexts” (Savage 1954, p. 80), seemingly not realizing, at this point, that his postulates imply bounded utility.
Suppose \(\eta ( 0 )>0\). Then by continuity of \(\eta ({\bullet })\), there exists an interval \([0,\bar{c}]\) on which \(\eta ( c )>\varepsilon >0\). Now note that
$$\begin{aligned} -\log u^{{\prime }}( \bar{c} )+\log u^{{\prime }}( \underline{c} )&= \int \limits _{\underline{c}}^{\bar{c}} \rho ( c )dc\\&= \int \limits _{\underline{c} }^{\bar{c}} \left[ {\frac{\eta ( c)}{c}} \right] dc \ge \varepsilon \int \limits _{\underline{c} }^{\bar{c}} \frac{1}{c}dc \end{aligned}$$so that \(u^{{\prime }}( {\underline{c}} )\ge u^{\prime }(\bar{c})(\bar{c}/\underline{c})^{\varepsilon }\). Hence when taking \({\underline{c}} \rightarrow 0\) on both sides, we find \(\lim _{c\rightarrow 0} u^{{\prime }} (c )=\infty \). To show that the converse is not true, consider a utility function that coincides with \(u( c)= c(1-\log (c))\) on some interval \([ {0,\bar{c}}],\) where \(\bar{c} \le 1\). Then \(u^{{\prime }}(c)=-\log (c)\), so \(\lim _{c\rightarrow 0} u^{{\prime }}(c)=\infty \). However, \(\eta ( c )=-1/\log (c)\), so \({\lim }_{c\rightarrow 0} \eta ( c )=0\). All we can say is that if \({\lim }_{c\rightarrow 0} u^{{\prime }}( c )= \infty \) and \(\eta (c)\) converges to \(0\) as \(c\rightarrow 0\), it must be doing so at a rate more slowly than any power of \(c\).
At the time of writing, the latest annual data available was for 2009.
An increase of this magnitude corresponds to the upper limit of what the International Panel on Climate Change considers a likely range for the following century.
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We are grateful to Matthew Agarwala (CSERGE, UEA) for assistance in formatting elements of this paper.
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The analysis and conclusions set forth in this paper are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors of the Federal Reserve System.
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Arrow, K.J., Priebsch, M. Bliss, Catastrophe, and Rational Policy. Environ Resource Econ 58, 491–509 (2014). https://doi.org/10.1007/s10640-014-9788-6
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DOI: https://doi.org/10.1007/s10640-014-9788-6