Linear-quadratic approximation of optimal policy problems

We consider a general class of nonlinear optimal policy problems with forward-looking constraints, and show how to derive a problem with linear constraints and a quadratic objective that approximates the exact problem. The solution to the LQ approximate problem represents a local linear approximation to optimal policy from the “timeless perspective” proposed in Benigno and Woodford (2004, 2005) [6,7], in the case of small enough stochastic disturbances. We also derive the second-order conditions for the LQ problem to have a solution, and show how to correctly rank alternative simple policy rules, again in the case of small enough shocks.