Comparative Risk Aversion under Background Risks Revisited

This note determines a sufficient condition on (von Neumann-Morgenstern) utility functions to preserve (reserve) comparative risk aversion under general background risks. Our condition is weaker than the one determined by Nachman (1982, Journal of Economic Theory). Nachmanfs condition requires the monotonicity in the global sense, in other hand our condition only requires it in the local sense. And this generalization may make the condition on utility functions to hold the desirable property consisitent with the recent empirical observation.