A Simple Algorithm for Solving Ramsey Optimal Policy with Exogenous Forcing Variables

TL;DR

This paper introduces a straightforward algorithm for solving Ramsey optimal policy problems in DSGE models with exogenous forcing variables, extending existing methods by incorporating an additional Sylvester equation step and optimal anchoring.

Contribution

It extends Ljungqvist and Sargent's algorithm to include exogenous forcing variables and provides a practical implementation using standard matrix routines in Matlab and Scilab.

Findings

Efficient algorithm for Ramsey policy with exogenous variables

Incorporates Sylvester equation step for improved solutions

Enables computation of VAR representations with policy rules

Abstract

This algorithm extends Ljungqvist and Sargent (2012) algorithm of Stackelberg dynamic game to the case of dynamic stochastic general equilibrium models including exogenous forcing variables. It is based Anderson, Hansen, McGrattan, Sargent (1996) discounted augmented linear quadratic regulator. It adds an intermediate step in solving a Sylvester equation. Forward-looking variables are also optimally anchored on forcing variables. This simple algorithm calls for already programmed routines for Ricatti, Sylvester and Inverse matrix in Matlab and Scilab. A final step using a change of basis vector computes a vector auto regressive representation including Ramsey optimal policy rule function of lagged observable variables, when the exogenous forcing variables are not observable.