Robust comparative statics of risk changes

The standard method for establishing the comparative statics of risk changes in optimization problems has been confined to comparing unique interior solutions, relying on strong assumptions about payoff functions and decision variables. We propose a simple and intuitive approach that hinges on considerably weaker assumptions. Merging insights from the monotone comparative statics literature with insights from the risk apportionment literature, we show that the ranking of simple lottery pairs is all that is needed for establishing the comparative statics of risk changes. We use this approach to analyze the comparative statics of Nth-degree stochastic dominance shifts in a general setting with one and with multiple decision variables, and we show how these results can be applied to generalize the classical theories of precautionary saving, self-protection, and others.