Puzzles and the Behavioural New Keynesian Model
October 5, 2017
Herein I conduct similar policy experiments to those of Carlstrom et al. (2012) with a Behavioural New Keynesian (BNK) model, instead of the standard New Keynesian model (only) or a sticky information model. The BNK model does a good job of solving previous issues with the standard model (as already noted by Gabaix).
All of this work is already, explicitly or implicitly, in the cited articles, so I wouldn’t claim to show anything new. Nontheless, replication is a good way to learn.
The Model
We will take up the BNK model of Gabaix (2016):
\[ x_t = M E_t\{ x_{t+1} \} - \sigma (i_t - E_t\{ \pi_{t+1} \} - r_t^n) \hspace{0.3cm} [1] \]
\[ \pi_t = \beta M^f E_t\{ \pi_{t+1} \} + \kappa x_t \hspace{0.3cm} [2] \]
Forward Guidance
Following Carlstrom et al. (2012) and (2015), assume that at \(t > N+T\) the monetary authority follows an interest rate rule such as:
\[ i_t = r_t^n + \phi_\pi \pi_t \hspace{0.3cm} [3] \]
where the Taylor principle holds (\(\phi_\pi > 1\)), so that the system characterized by \([1]-[2]-[3]\) has the unique solution \(x_t = \pi_t = 0 \ \forall t > N+T\); under perfect foresight (\(E_t( g_{t+k}) - g_{t+k} = 0 \ \forall k \in \mathbb{Z}\) where \(g\) represents any variable). Now, consider a shock to the natural rate of interest that triggers \(i_t = 0\) for \(t = 0, 1, 2,..., N\); where the ZLB binds. The shock occurs at \(t=0\) where \(z_t \equiv i_t - r_t^n = -r_t^n > 0\) for \(t = 0, 1, 2,..., N\). This is followed by a stimulative rate peg \(z_t = i^* \leq 0\) for \(t = N+1,N+2,..., N+T\). We can rewrite the system characterized by \([1]\) and \([2]\) and \(z_t\); and use the unique solution for \(t>N+T\) as a terminal condition to ensure equilibrium uniqueness along the entire path while we run time backwards from \(T+1\).
During this period, inflation dynamics are governed by:
\[ \textbf{s}_{N+T-k} = \textbf{A}_1 \textbf{s}_{N+T-k+1} - (\kappa \sigma, 0)^T z_{N+T-k} \hspace{0.3cm} [4] \]
for \(k = 1, 2,...\)
where:
\[ \textbf{s}_{N+T-k} = ( \pi_{N+T-k}, \pi_{N+T-k+1} )^{T} \]
and
\[ \textbf{A}_1 = \begin{bmatrix} M + \kappa\sigma + \beta M^f & -M\beta M^f \\ 1 & 0 \end{bmatrix} \]
The forward guidance puzzle arrises from the explosive behaviour of the system described in \([4]\). The system is stable only if \(\textbf{A}_1\) has both eigenvalues inside the unit circle. This is the case when the strong rationality condition holds:
\[ (1-\beta M^f)(1-M) < \kappa\sigma \]
We can also see the behaviour of the system during a zero lower bound (ZLB) episode, where \(\kappa = \sigma = 0.5\), \(\beta = 0.95\), \(M^f = 1\) (firms are rational) and the real rate is \(-0.01\), \(4\%\) anually, for 8 periods. For a high enough consumer myopia the system seems to be stable. The results are highly sensitive to the values of \(\kappa \sigma\) chosen, as implied by the previous condition.
We can now conduct an experiment similar to that of Calstrom et al. (2012); where we mantain \(z_t\) a slightly negative value to simulate a stimulative peg at 0 with an assumed nominal steady state interest rate of \(4\%\), so: \(i^* = -0.01\) for 8 periods. We can now plot inflation at \(t=0\) as a function of \(T\).
This last figure is “my” equivalent of Figure 1 in Calstrom et al. (2012) [note the different parameter values for \(\kappa\) and \(\sigma\) lead to such high values, also my \(\sigma\) denotes a different constant than Calstrom et al.’s \(\sigma\)]. As we can see, little or no bias (black and red) get us an explosive behaviour, while a strong enough bias (blue and green) deliver stable dynamics.
References
Carlstrom, Charles T., Timothy S. Fuerst, and Matthias Paustian. “Inflation and output in New Keynesian models with a transient interest rate peg.” (2012).
Gabaix, Xavier. “A Behavioral New Keynesian Model.” (2017).