First-order risk aversion and non-differentiability

Summary First-order risk aversion happens when the risk premiump a decision maker is willing to pay to avoid the lottery$$t \cdot \tilde \varepsilon , E[\tilde \varepsilon ] = 0$$, is proportional, for smallt, tot. Equivalently,$$\partial \pi /\partial t|_{ t = 0^ + } > 0$$. We show that first-order risk aversion is equivalent to a certain non-differentiability of some of the local utility functions (Machina [7]).