# Hybrid Phillips curve under two-state Markov process

I am studying a New Keynesian model with a hybrid Phillips curve which reads \begin{align*} \pi_t=\beta E_t \pi_{t+1} + \varpi \pi_{t-1}+ \kappa x_t. \end{align*} The demand shock in the IS curve follows a two-state Markov process, i.e. I assume that there are two states, states $$n$$ and $$z$$. The transition probabilities are given as follows \begin{align*} P(s_{t+1}=n| s_t=n)&=p\\ P(s_{t+1}=z| s_t=n)&=1-p\\ P(s_{t+1}=z| s_t=z)&=q\\ P(s_{t+1}=n| s_t=z)&=1-q\\ \end{align*} For $$\varpi=0$$, it follows that the Phillips curve in state $$n$$ reads \begin{align*} \pi_n = \beta\left[ p\pi_n + (1-p)\pi_{z}\right]+ \kappa x_n. \end{align*} However, I am interested in the Phillips curve in state $$n$$ for the case where $$\varpi\neq 0$$. My idea is that, given the economy is in state $$n$$ today, this can be only because

1. The economy was in state $$n$$ in $$t-1$$ and remained in state $$n$$ with probability $$p$$, or
2. The economy was in state $$z$$ in $$t-1$$ and switched to state $$n$$ with probability $$1-q$$.

Following this idea, for $$\varpi\neq 0$$, I would write down the Phillips curve in state $$n$$ as \begin{align*} \pi_n = \beta\left[ p\pi_n + (1-p)\pi_{z}\right]+ \varpi\left[p \pi_n + (1-q)\pi_z\right]+ \kappa \pi_n. \end{align*} Is this approach correct? If not, how do I add $$\pi_{t-1}$$ given the economy is in state $$n$$ in period $$t$$?

• Hi: what does $x_t$ represent ? is that the demand shock ? Feb 17, 2021 at 17:51
• No, x_n is the output gap which is determined via another equation (IS curve). I left it out as it (in my view) does not matter for the question at hand, although the shock r_t appears in the IS curve. Feb 17, 2021 at 18:07
• but where did you get the $\kappa \pi_n$ as the last term ? Feb 18, 2021 at 4:16