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Abstract
We investigate the impact of a non-financial background risk 𝜀̃ on the
preference rankings between two independent financial risks 1 and
2 for an
expected-utility maximizer. More precisely, we provide necessary and sufficient
conditions for the alternative (x0+
1,y0+ 𝜀̃) to be preferred to
(x0+
2,y0+ 𝜀̃)
whenever (x0+
1,y0) is
preferred to (x0+
2,y0). Utility
functions that preserve the preference rankings are fully characterized. Their
practical relevance is discussed in light of recent results on the constraints
for the modelling of the preference for the disaggregation of harms.
Acknowledgements
The authors acknowledge helpful discussions and exchanges with L. Eeckhoudt, H. Schlesinger and B. Versaevel. All remaining errors are ours.
Notes
1 See also CitationKeeney and Raiffa (1976), which consider related utility independence conditions.
2 The fact that any additive utility function U(x,y)=ux(x)+uy(y) preserves preference rankings on the first argument in the presence of any background risk on the second argument, irrespective of their dependence, is obvious. The necessity almost straightforwardly results from the considerations of CitationPollak (1967) on the so-called k-standard lottery tickets for which the preservation of preference rankings must hold.
3 Here, it would be better to write U(x,y)=u(x·(1+y)) with E[y]=0 as an actuarially neutral multiplicative background risk must have a mean value equal to one instead of zero (see the discussion in CitationFranke et al, 2006), but this shift in the expectation does not change our conclusions.
4 This concept was first introduced by CitationRichard (1975), and was explored further by CitationEpstein and Tanny (1980).