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Published in: Theory and Decision 1/2020

25-01-2020

Some conditions for the equivalence between risk aversion, prudence and temperance

Authors: Marzia De Donno, Mario Menegatti

Published in: Theory and Decision | Issue 1/2020

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Abstract

We study relationships between different aspects of risk preferences. We show that, under the assumptions of non-satiation and bounded marginal utility, some additional conditions on the asymptotic behaviour of the indices of relative prudence and relative temperance ensure that risk aversion, prudence and temperance are equivalent. Similar conclusions are derived for higher-degree risk aversion. Moreover, some links between indices of relative risk aversion of different degrees are derived. The implications of these results for several economic problems which involve risk changes are discussed.

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Appendix
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Footnotes
1
De Donno et al. (2019) derived, in the context of a portfolio problem, some restrictive conditions which ensure that the threshold condition on the index of partial relative risk aversion of some order implies the threshold conditions on the same indices of lower orders. We also recall that Maggi et al. (2006) studied the linkages between the monotonicity of the index of relative risk aversion and the index of relative prudence.
 
2
These are just some of the possible applications of our results, which can generally be used in all problems where changes in risks are relevant, such as production localization problems (as in Eeckhoudt et al. 2009) labour supply choice (as in Chiu and Eeckhoudt 2010), optimal self protection (as in Crainich et al. 2016).
 
3
Throughout the paper, we say that the limit of a function f as x approaches infinity exists when \(\lim _{x\rightarrow +\infty } f(x) = l \in {\mathbb {R}}\) or \(\lim _{x\rightarrow +\infty } f(x) = \pm \infty\)
 
4
Consider as an example for the risk averse case the classical CRRA utility and for the risk lover case the utility function \(U(x)=x\;\arctan (x)-\log (1+x^2)\) on the domain \([0, +\infty )\).
 
5
Many economic models suggest that this additional utility should decrease when wealth increases. This is what is called the “Law of diminishing marginal utility” (e.g. Marshall 1920, p.79). This idea, however, implies risk aversion, which is clearly an assumption we do not introduce in our paper since risk aversion is one of the aspects of preferences involved in our results.
 
6
All rational functions composed of exponentials, logarithmic, polynomial functions or the inverse tangent have a limit at \(+\infty\) (which can be infinite).
 
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Metadata
Title
Some conditions for the equivalence between risk aversion, prudence and temperance
Authors
Marzia De Donno
Mario Menegatti
Publication date
25-01-2020
Publisher
Springer US
Published in
Theory and Decision / Issue 1/2020
Print ISSN: 0040-5833
Electronic ISSN: 1573-7187
DOI
https://doi.org/10.1007/s11238-020-09745-5

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