Weak proper risk aversion and the tempering effect of background risk.

Summary In this paper we study a natural restriction on utility functions. More specifically, we look for utility functions such that any undesirable risk can never become desirable by the introduction of another independent risk of negative or zero expectation. This condition - called "weakly prope" - generalizes the concept of "properness" and "standardness". An important property of this family of utility functions is that the introduction of an underlying risk to an agent's wealth necessarily makes this agent more riscophobic. We present the necessary and sufficient condition to describe this family. The set of utility functions with decreasing and convex absolute aversion belongs to this family.