Preferences of economic agents may evolve over time due to learning, changing attitudes toward risk, or the incorporation of additional information. Motivated by this observation, we develop second-order envelope theorems for utility maximization under small perturbations of preferences. As an application, we introduce a framework for the Greeks of indifference prices, describing how these prices respond to evolving preferences.
Our approach leads to a number of auxiliary results of independent interest. In particular, we obtain second-order expansions of the primal and dual value functions and first-order expansions of the corresponding optimal wealth and supermartingale deflator. The expansion coefficients are characterized by quadratic optimization problems and possess a transparent representation in terms of orthogonal projections under the risk-tolerance measure when the latter exists.
The resulting formulas suggest that the first-order sensitivities of indifference prices are determined exclusively by the nonhedgeable component of the preference perturbation under the optimal numéraire.
