Analyzing Linear DSGE models: the Method of Undetermined Markov States

TL;DR

This paper introduces a novel analytical method for solving linear DSGE models with a single endogenous state variable by representing them as a three-state Markov chain, enabling closed-form solutions for key economic metrics.

Contribution

The paper develops a new analytical approach using Markov chain representation for linear DSGE models, simplifying computations of impulse responses and other objects.

Findings

Closed-form solutions for impulse responses and present value multipliers

Efficient analytical method reduces reliance on numerical solutions

Application to a New Keynesian model with optimal monetary policy

Abstract

I show that a class of Linear DSGE models with one endogenous state variable can be represented as a three-state Markov chain. I develop a new analytical solution method based on this representation, which amounts to solving for a vector of Markov states and one transition probability. These two objects constitute sufficient statistics to compute in closed form objects that have routinely been computed numerically: impulse response function, cumulative sum, present discount value multiplier. I apply the method to a standard New Keynesian model that features optimal monetary policy with commitment.