Hopf Bifurcation From New-Keynesian Taylor Rule to Ramsey Optimal Policy

Abstract

This paper compares different implementations of monetary policy in a new-Keynesian setting. We can show that a shift from Ramsey optimal policy under short-term commitment (based on a negative feedback mechanism) to a Taylor rule (based on a positive feedback mechanism) corresponds to a Hopf bifurcation with opposite policy advice and a change of the dynamic properties. This bifurcation occurs because of the ad hoc assumption that interest rate is a forward-looking variable when policy targets (inflation and output gap) are forward-looking variables in the new-Keynesian theory.JEL classification numbers: C61, C62, E43, E44, E47, E52, E58.

Imperfect Credibility Versus No Credibility of Optimal Monetary Policy

“…Hence, Blanchard and Kahn (1980) determinacy condition implies more stable eigenvalues with limited credibility model with respect to zero credibility. For example, including the private sector consumption Euler equation besides the new-Keynesian Phillips curve leads to a Hopf bifurcation (Chatelain and Ralf (2017b)). This also implies that the number of non-collinear variables in a vector auto-regressive representation is smaller with zero credibility than with limited credibility (Chatelain and Ralf (2017c)).…”

“…Ljungqvist and Sargent (2012, chapter 19) seek the stationary equilibrium process using the augmented discounted linear quadratic regulator solution of the Hamiltonian system (Hansen and Sargent (2007)) as an intermediate step (Chatelain and Ralf (2017) algorithm). This amount to seek a stable subspace of dimension two in a system of three equations including the marginal condition on the policy instrument (or on the Lagrange multiplier on in ‡ation).…”

Ramsey Optimal Policy in the New-Keynesian Model With Public Debt

“…(3) In the new-Keynesian model, inflation and output-gap dynamics do not depend on public-debt dynamics. For this reason, if the volatility of public debt has a zero weight in the welfare and policy maker's loss function, the policy implications of Ramsey optimal policy for the Taylor rule are exactly the same as in a model without public debt (Chatelain and Ralf (2020a)). In addition, a surplus rule stabilizes public-debt dynamics.…”

“…We compute Ramsey optimal policy in a discrete-time new-Keynesian model with public debt. We compare it with Woodford (1996) and Cochrane (2019) approaches with simple rules and with Chatelain and Ralf (2020a) Ramsey optimal policy without debt. We get the following policy implications:…”

Ramsey Optimal Policy in the New-Keynesian Model with Public Debt

“…The optimal Taylor rule parameter for in ‡ation is larger than one (instead of below one for the FTPL). The optimal Taylor rule parameter for output gap is negative, because of the intertemporal substitution e¤ect of interest rate on output gap (Chatelain and Ralf (2019b)). Both Taylor rule optimal parameters implies the local stability of in ‡ation and output gap dynamics.…”