Considering the New Keynesian Model we have the Phillips curve and dynamic IS curve in log-linearized form with price shock $u^{\pi}$ and demand shock $u^{IS}$ :$$\pi_t=\beta E_t\pi_{t+1}+\kappa(y_t-y_t^n)+u_t^{\pi}$$ $$y_t=E_{t+1}-\frac{1}{\sigma}(i_t-E_t\pi_{t+1}-\rho)+u_t^{IS}$$

where $\kappa>0$, $1>\beta>0$,$\sigma>0$. The shocks follow some stochastic process.Furthermore, natural level of output $y_t^n$ is determined by a technology shock, also following a stochastic process.

Now, we have the following Taylor rule of the central bank: $$i_t=\rho+\phi_{\pi}E_t\pi_{t+1}+\phi_y E_t(y_{t+1}-y_{t+1}^n).$$

I know that following this Taylor rule, we are not able to stabilize any of these shocks. However, could someone tell me step by step how me come to this conclusion? Also does our conclusion change if we would use t terms rather than expected (t+1) ones?

I came so far:

1. Price shock: inflation increases, central bank reacts (because of higher $E_t\pi_{t+1}$?) with higher interest rate.And higher interest rate leads to lower demand and production($y_t$). Since in order to decrease inflation an output gap has to be created, divine coincidence does not hold and stabilization is not possible. This is actually a reasoning that my teacher gave. However, if output gap exist because of an attempt to lower inflation, could the inflation gap not close after inflation is stable again? Also, as before, I do not know if we can use expected terms like actual terms. Also, if it is stated that shocks follow a stochastic process, actually all expected terms for the next period, shouldn't they be all 0?

2. In terms of a demand shock, demand increases leading to an increase in actual output, creating a positive output gap, leading to an increase in inflation. If my understanding of the expected terms is right, central bank would expect positive inflation also for the next period? And hence interest rate increases (also with increase of expected output gap)? Then demand decreases and output gap decreases. My teacher stated in the solutions that at some point the interest rate adjusts for the shock, however, not totally. I do not see this though. Could someone explain to me the obvious?

3.In terms of a technology shock, a negative output gap leads to a decrease in inflation. Because of the negative output gap, also a negative output gap should be expected in the future (if we assumed an AR process?), that comes with decrease in inflation, hence, central bank lowers interest rate to adjust demand and output.

I would appreciated it if someone could tell me step by step how the reasoning is done here to come to the conclusion that this Taylor rule cannot stabilize inflation.


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