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This paper determines a new sufficient condition of the (von Neumann-Morgenstern) utility function that preserves comparative risk aversion under background risk. It is the single crossing condition of risk aversion. Because this condition requires monotonicity in the local sense, it may satisfy the U-shaped risk aversion observed in the recent empirical literature.

Pratt [

Important contributions to this question are from Kihlstrom et al. [

The organization of the paper is as follows. In Section

Because our setting is basically identical to that of Nachman [

Let us define the derived utility function as

Let us define the function

Pratt [

Before providing the theorem, we define the notion of the single crossing condition:

Suppose that

For the preparation of the proof, we provide the following two lemmas. A similar result to the first lemma was obtained by Osaki [

The proof of Lemma

Lemma _{2} property.) If we let

Let us assume that

In this subsection, we provide a proof of Theorem

From Lemma

We close this section with a comment on the relationship between our analysis and Nachman's [

(see, e.g., Gollier [

Because monotone functions imply functions that satisfy the single crossing condition, the condition determined in this paper is weaker than that determined by Nachman [

In this section, we consider two specific forms of background risk: additive and multiplicative. We specify additive background risk which has the additive payoff function

Over the past three decades, many studies considered the effects of additive background risk, which has the additive payoff function,

We define

Suppose that

In a recent paper, Franke et al. [

Suppose that

Monotone functions are functions that satisfy the single crossing condition. Hence, Corollaries

they preserve comparative risk aversion under additive (multiplicative) background risk;

they are consistent with recent empirical findings, for example, absolute (relative) risk aversion is decreasing in low wealth and increasing in high wealth.

In this paper, we determined a new sufficient condition for utility functions that preserve comparative risk aversion under background risk. The condition determined by Kihlstrom et al. [

It follows from a straightforward calculation that

The authors would like to thank Mark Bremer, Chiaki Hara, John Quiggin Katsushige Sawaki, and especially an anonymous referee for their useful comments and constructive suggestions. This research was partly supported by Grants-in-Aid for JSPS Fellows (no. 02205), Grants-in-Aid for Research Activity Start-up (no. 21830146), the Japan Securities Scholarship Foundation, and Nihon Housei Gakkai Foundation. Of course, all remaining errors are the authors.