Interest rate rule in monetary DSGE model

Question

I am studying DSGE models and try to solve exercise 2.2 from Gali's (2008) book. In short, consider the simple classical economy where the following approximate equilibrium conditions must be satisfied: $$y_t = E_t\{y_{t+1}\} - \frac{1}{\sigma}(i_t - E_t\{\pi_{t+1}\} - \rho)$$ and $$r_t = i_t - E_t\{\pi_{t+1}\}$$ and $y_t$ and $r_t$ determined independently of monetary policy.

Derive an interest rate rule that guarantees full stabilization of inflation (i.e. $\pi_t = \pi^* \;\; \forall t$).

I thought of the simple rule like $i_t = \phi\pi_t$. It is possible to show (using forward iteration) that in this case $$\pi_t = \sum_{i=0}^{\infty}\frac{1}{\phi^{i+1}}E_t\{r_{t+i}\}$$ and letting $\phi \to \infty$ inflation goes to zero. However, I think that this not what was meant by the author. Any help appreciated.

Answer 1

I've just solved this problem. First of all, your solution does not make too much sense, as in a simple interest rate rule it must hold that the sum of all coefficients must be greater than one. In your case this means that $\phi>1$. Therefore, the series would converge not to zero. Second, an interest rate rule should try to offset fluctuations. This means that the central bank should try to offset gaps from its target (like in the task). Try instead the following interest rate rule. \begin{equation} i_t=\rho+\phi_{\pi}(\pi_t-\pi)+\sigma(\Delta y_{t+1})+\pi \end{equation} Now what is the idea? Assmung that the output does not change, this term is zero. Using the Fisherian $i_t=r_t+\mathbb{E}_t\{\pi_{t+1}\}$ we get \begin{equation} \rho+\phi_{\pi}(\pi_t-\pi)+\sigma(\Delta y_{t+1})+\pi \overset{!}{=}r_t+\mathbb{E}_t\{\pi_{t+1}\} \end{equation} which can be summarized as \begin{equation} \phi_{\pi}^{-1}\left(r_t-\rho+\mathbb{E}_t{\bar{\pi}_{t+1}}-\sigma(\Delta y_{t+1})\right)=\bar{\pi}_t, \end{equation} where $\bar{\pi}_t=\pi_t-\pi$. Solving forward yields \begin{equation} \bar{\pi}_t=\sum_{k=0}^{\infty}\phi_{\pi}^{-(k+1)}\mathbb{E}_t\{r_{t+k}-\rho-\sigma\Delta y_{t+1+k}\} \end{equation} Notice that for $|\phi_{\pi}|>1$, the series converges. However, your log-linearized Euler equation can be rearranged such that it holds that $r_t=\rho+\sigma\mathbb{E}_t{\Delta y_{t+1}}$. Finally, this can be plugged into $\bar{\pi}_t$ such that $\bar{\pi}_t=0 \quad \forall t$. Notice that this expression finally is your answer, as $\bar{\pi}_t=\pi_t-\pi=0$ means that the central bank always achieves its target.