Risk and Monotone Comparative Statics without Independence
Collin Raymond, Yangwei Song
TL;DR
The paper develops a unified framework for monotone comparative statics under non-EU preferences by replacing the Bernoulli utility with local utility functions $u(z,F)$ and employing differentiability concepts such as Hadamard and Gateaux. It establishes three main sufficiency conditions—SC1, SC2, and supermodularity—for MCS when actions affect distributions and parameters affect utilities, and shows how these extend or nest EU results, enabling tractable analysis in portfolio choice and precautionary savings. The authors provide motivating examples (including DARA and precautionary motives) to illustrate the limits of naively transferring EU conditions to non-EU settings and demonstrate that non-EU models can yield qualitatively different comparative statics. The framework is applied to three key problems (risk-based portfolio choices, wealth-driven risk-taking, and precautionary saving), with corollaries for RDU and Kreps–Porteus representations, and the paper discusses extensions to ambiguity and multi-stage settings via extensive appendices. Overall, the work broadens the practical toolkit for robust comparative statics in economics and finance beyond the confines of EU models, enabling precise predictions under a wide class of non-EU preferences.
Abstract
We extend well-known comparative results under expected utility to models of non-expected utility by providing novel conditions on local utility functions. We illustrate how our results parallel, and are distinct from, existing results for monotone comparative statics under expected utility, as well as risk preferences for non-expected utility. Our conditions generalize existing results for specific preferences (including expected utility) and allow us to verify monotone comparative statics for novel environments and preferences. We apply our results to portfolio choice problems where preferences or wealth might change, as well as precautionary savings.