Self-protection under Nth-degree risk increase of random unit cost

Cost risk, as a type of multiplicative risk, should be given more attention in decision-making issues. Crainich and Menegatti (2021) have studied the effects of introducing random unit cost in self-protection under the four standard self-protection model frameworks. They focus on the case where the unit cost of effort in self-protection changes from certainty (denoted as $c$) to randomness (denoted as $\tilde{c}$) with $E\left[\tilde{c}\right]=c$, which represents second-degree risk increase in Ekern (1980). In this paper, we generalize the concept of second-degree risk increase to Nth-degree risk increase and provide sufficient conditions for increasing or decreasing effort in self-protection, which are closely related to the parity of the order of the risk change and decision-maker's higher-order risk attitudes. We use the multiplicative effect and apportionment effect to explain the decision-maker's preference conditions.