<p>I am studying a New Keynesian model with a hybrid Phillips curve which reads
<span class="math-container">\begin{align*}
\pi_t=\beta E_t \pi_{t+1} + \varpi \pi_{t-1}+ \kappa x_t.
\end{align*}</span>
The demand shock in the IS curve follows a two-state Markov process, i.e. I assume that there are two states, states <span class="math-container">$n$</span> and <span class="math-container">$z$</span>. The transition probabilities are given as follows
<span class="math-container">\begin{align*}
P(s_{t+1}=n| s_t=n)&amp;=p\\
P(s_{t+1}=z| s_t=n)&amp;=1-p\\
P(s_{t+1}=z| s_t=z)&amp;=q\\
P(s_{t+1}=n| s_t=z)&amp;=1-q\\
\end{align*}</span>
For <span class="math-container">$\varpi=0$</span>, it follows that the Phillips curve in state <span class="math-container">$n$</span> reads
<span class="math-container">\begin{align*} 
\pi_n = \beta\left[ p\pi_n + (1-p)\pi_{z}\right]+ \kappa x_n.
\end{align*}</span>
However, I am interested in the Phillips curve in state <span class="math-container">$n$</span> for the case where <span class="math-container">$\varpi\neq 0$</span>. My idea is that, given the economy is in state <span class="math-container">$n$</span> today, this can be only because</p>
<ol>
<li>The economy was in state <span class="math-container">$n$</span> in <span class="math-container">$t-1$</span> and remained in state <span class="math-container">$n$</span> with probability <span class="math-container">$p$</span>, or</li>
<li>The economy was in state <span class="math-container">$z$</span> in <span class="math-container">$t-1$</span> and switched to state <span class="math-container">$n$</span> with probability <span class="math-container">$1-q$</span>.</li>
</ol>
<p>Following this idea, for <span class="math-container">$\varpi\neq 0$</span>, I would write down the Phillips curve in state <span class="math-container">$n$</span> as
<span class="math-container">\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*}</span>
Is this approach correct? If not, how do I add <span class="math-container">$\pi_{t-1}$</span> given the economy is in state <span class="math-container">$n$</span> in period <span class="math-container">$t$</span>?</p>

CC BY-SA 4.0
2021-02-17T17:51:39.530Z
Hi: what does $x_t$ represent ? is that the demand shock ?
mark leeds
CC BY-SA 4.0
2021-02-17T18:07:54.157Z
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.
chopschoc
CC BY-SA 4.0
2021-02-18T04:16:57.393Z
but where did you get the $\kappa \pi_n$ as the last term ?
mark leeds
<new-keynesian-economics><markov-chain>
Hybrid Phillips curve under two-state Markov process
chopschoc
# Hybrid Phillips curve under two-state Markov process

(Asked by: username_0 on 2021-02-16)

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 $t$?

**Tags:** new-keynesian-economics, markov-chain

> **Comments:**
> **username_1**: Hi: what does $x_t$ represent ? is that the demand shock ?  
> **username_0**: 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.  
> **username_1**: but where did you get the $\kappa \pi_n$ as the last term ?  

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## 0 answers

