##
###
A Toolkit for Solving Models with a Lower Bound on Interest Rates of Stochastic Duration
[report]

Gauti Eggertsson, Sergey Egiev, Alessandro Lin, Josef Platzer, Luca Riva

2020
unpublished

This paper presents a toolkit to solve for equilibrium in economies with the effective lower bound (ELB) on the nominal interest rate in a computationally efficient way under a special assumption about the underlying shock process, a two-state Markov process with an absorbing state. We illustrate the algorithm in the canonical New Keynesian model, replicating the optimal monetary policy in Eggertsson and Woodford (2003) , as well as showing how the toolkit can be used to analyse the
## more »

... yse the medium-scale DSGE model developed by the Federal Reserve Bank of New York. As an application, we show how various policy rules perform relative to the optimal commitment equilibrium. A key conclusion is that previously suggested strategies -such as price level targeting and nominal GDP targeting -do not perform well when there is a small drop in the price level, as observed during the Great Recession, because they do not imply a sufficiently strong commitment to low future interest rates ("make-up strategy"). We propose two new policy rules, Cumulative Nominal GDP Targeting Rule and Symmetric Dual-Objective Targeting Rule that are more robust. Had these policies been in place in 2008, they would have reduced the output contraction by approximately 80 percent. If the Federal Reserve had followed Average Inflation Targeting -which can arguably approximate the new policy framework announced in August 2020 -the output contraction would have been roughly 25 percent smaller. Abstract This paper presents a toolkit to solve for equilibrium in economies with the effective lower bound (ELB) on the nominal interest rate in a computationally efficient way under a special assumption about the underlying shock process, a two-state Markov process with an absorbing state. We illustrate the algorithm in the canonical New Keynesian model, replicating the optimal monetary policy in Eggertsson and Woodford (2003) , as well as showing how the toolkit can be used to analyse the medium-scale DSGE model developed by the Federal Reserve Bank of New York. As an application, we show how various policy rules perform relative to the optimal commitment equilibrium. A key conclusion is that previously suggested strategies -such as price level targeting and nominal GDP targeting -do not perform well when there is a small drop in the price level, as observed during the Great Recession, because they do not imply a sufficiently strong commitment to low future interest rates ("make-up strategy"). We propose two new policy rules, Cumulative Nominal GDP Targeting Rule and Symmetric Dual-Objective Targeting Rule that are more robust. Had these policies been in place in 2008, they would have reduced the output contraction by approximately 80 percent. If the Federal Reserve had followed Average Inflation Targeting -which can arguably approximate the new policy framework announced in August 2020 -the output contraction would have been roughly 25 percent smaller. The effective lower bound (ELB) on nominal interest rates has been widely studied in recent years. It is standard to analyse this problem with dynamic stochastic general equilibrium (DSGE) models, where the ELB shows up as an inequality constraint on the nominal interest rate. However, inequality constraints complicate the application of standard solution strategies, e.g. perturbation methods. These methods approximate the behaviour of a dynamical non-linear model around a point (usually, but not necessarily, via linearisation) using differentiability assumptions. Occasionally binding constraints pose a challenge for direct application of these methods. In this paper, we present a toolkit aiming to facilitate the application of a generalised version of the solution method first used by Eggertsson and Woodford (2003) , who analyse the ELB in the face of a two-state Markov process for the exogenous shocks with an absorbing state. 1 We illustrate the algorithm in the canonical New Keynesian (NK) model and in the medium-scale DSGE model developed by the Federal Reserve Bank of New York (FRBNY). As an economic application, we consider various policy rules and study their performance relative to the optimal commitment equilibrium. Previously suggested policy rules -such as price level targeting and nominal GDP targeting -do not perform well when the price level does not fall by a large amount, as observed during the Great Recession, because they do not imply sufficiently strong commitment to low future interest rate ("make-up strategy"). This also applies to a policy rule we term Average Inflation Targeting, which arguably approximates the new policy regime of the Federal Reserve recently presented by Powell (2020). To solve this shortcoming, we propose two new policy rules, a Cumulative Nominal GDP Targeting Rule and a Symmetric Dual-Objective Targeting Rule that are more robust. Had either of these policy rules been in place in 2008, and believed to be credible, the model simulation suggests the Federal Reserve would have reduced the output contraction (relative to trend) by about 80-90 percent. The comparable number for the average inflation targeting rule is 25 percent (Table 3) . Several strategies have been proposed to deal with the presence of inequality constraints in DSGE models. Eggertsson and Woodford (2003) exploit a particular structure for the exogenous disturbances: the shock process implies that the model unexpectedly moves to a "crisis state" and then reverts back to the "steady state" with a fixed probability. Once back to the steady state, it stays there forever (i.e., the steady state is an absorbing state). The idea behind the approach is intuitive: instead of treating a single dynamical system that contains both a set of equality constraints and a set of occasionally binding inequality constraints, we split the system into several parts called regimes, each of which contains equality constraints exclusively. Once cast in this form, we can apply perturbation methods, since each equation is differentiable. An application to the ELB scenario should make this clear: we distinguish among four regimes, each of them corresponding to a different combination of the status of the inequality constraint (e.g. ELB binding or not) and the exogenous Markov disturbance (crisis or steady state). For the regimes that feature the ELB not binding, we treat the model as if the ELB was not present. In the other two regimes, when the ELB constraint is binding, the equilibrium conditions will be characterised by an equality constraint (e.g. i t = i t 1 = 0). Since all four dynamical systems are described by a set of equations, each can be solved using perturbation techniques. The assumptions on the shock structure allow us to solve the model recursively in regimes. Starting from 1 Source codes and examples are maintained at https://github.com/gautieggertsson/2-state-toolkit. Section A.2 in the Appendix presents a short user guide. Appendix B contains a number of illustrative examples.

doi:10.3386/w27878
fatcat:433d5ikheffdrp7oon5hjrfhce