The Family of Alpha,[a,b] Stochastic Orders: Risk vs. Expected Value

Abstract

In this paper we provide a novel family of stochastic orders, which we call the $\alpha,[a,b]$-concave stochastic orders, that generalizes second order stochastic dominance. These stochastic orders are generated by a novel set of "very" concave functions where $\alpha$ parameterizes the degree of concavity. The $\alpha,[a,b]$-concave stochastic orders allow us to derive novel comparative statics results for important applications in economics that could not be derived using previous stochastic orders. In particular, our comparative statics results are useful when an increase in the lottery's riskiness increases the agent's optimal action, but an increase in the lottery's expected value decreases the agent's optimal action. For this kind of situation, we provide a tool to determine which of these two forces dominates -- riskiness or expected value. We apply our results in consumption-savings problems, self-protection problems, and in a Bayesian game.