Perturbations in DSGE models: An odd derivatives theorem

Abstract

This paper extends a classical result in portfolio theory about the effect of risk on value functions to its effect more generally on policy functions. If odd moments of shocks are zero up to some order, then the odd order marginal effects on value and policy functions of introducing these shocks are zero as well. Mathematically, all coefficients of corresponding odd order in the perturbation parameter are zero. If shocks are symmetric, e.g. normally distributed, then this holds for all odd orders. The main theorem (1) generalizes past results on perturbations and unifies their economic intuition, (2) improves the computation of stochastic coefficients, and (3) illustrates how to derive properties of high order perturbations through simple induction.