Nonlinear risks: a unified framework

We study the conditions under which increasing risk raises the optimal control variable when the budget constraint is nonlinear. In contrast to the case when the budget constraint is linear, nonlinearities alter the risk attitudes of the economic agent; that is, the agent’s risk behavior is driven by the interaction between the shape of the utility function and the shape of the budget constraint. This paper complements the classical literature with linear payoff functions in two important ways. First, we derive necessary and sufficient conditions for unambiguous comparative statics of changes in risk when the budget constraint is nonlinear. Second, we highlight the critical role of the coefficients of technological improvement and cross-prudence in technology derived from the nonlinear budget constraint in a similar form to that in which the coefficient of relative risk aversion and the coefficient of relative prudence are derived from the utility function. We show several applications of this theory, including the worker’s decision to work in-person during a pandemic.