  In this paper we provide a novel family of stochastic orders that generalizes
second order stochastic dominance, which we call the $\alpha,[a,b]$-concave
stochastic orders.
  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 cannot be derived
using previous stochastic orders. In particular, our comparative statics
results are useful when an increase in a lottery's riskiness changes the
agent's optimal action in the opposite direction to an increase in the
lottery's expected value. 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.
