Weak Proper Risk Aversion And The Tempering Effect of Background Risk

We examine in this paper a new natural restriction on utility functions, namely that an undesirable risk can never be made desirable by the presence of an independent, unfair risk. This concept is called weak properness. It generalizes the concept of properness (individually undesirable, independent risks are always jointly undesirable) introduced by Pratt and Zeckhauser [1987]. An important property of weak properness and properness is that adding an unfair risk to wealth makes risk-averse people more risk averse, therefore increasing the equilibrium price of risk in exchange economies a la Lucas [1978]. Weak properness implies that the two first derivatives of the utility function are concave transformations of the original utility function. A sufficient condition for weak-properness is that absolute risk aversion be decreasing and convex.