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Log-linearizing optimal price in New Keynesian model

  1. Apr 22, 2013 #1
    1. The problem statement, all variables and given/known data
    I am going to do a log-linearization around a zero-inflation flexible price steady state of:

    [itex]\frac{P_{t}^{*}}{P_{t}}E_{t}\sum_{k=0}^{\infty}\theta^{k}\beta^{k}C_{t+k}^{1-\sigma}\left(\frac{P_{t+k}}{P_{t}}\right)^{\epsilon-1}[/itex]

    Zero-inflation flexible price steady state implies that [itex] P_{t} = P^{*}_{t} = P_{t+k} \equiv \bar{P} [/itex].


    3. The attempt at a solution

    I know the solution is:
    [itex]\frac{\bar{C}^{1-\sigma}}{1-\theta\beta} +\frac{\bar{C}^{1-\sigma}}{1-\theta\beta}(\hat{p}^{*}_{t} - \hat{p}_{t}) + \bar{C}^{1-\sigma}\sum_{k=0}^{\infty}\theta^{k}\beta^{k}[(1-\sigma)E_{t}\hat{c}_{t+k} + (\epsilon-1)(E_{t}\hat{p}_{t+k} - \hat{p}_{t})] [/itex]

    where [itex] \hat{x}_{t} = x_{t} - \bar{x}, x_{t} = \log(X_{t}), \bar{x} = \log(\bar{X}) [/itex] where [itex] \bar{X} [/itex] is the steady state value of X. And we have used [itex] \sum_{k=0}^{\infty}\theta^{k}\beta^{k} = \frac{1}{1-\theta\beta} [/itex].

    I've now tried for several hours to do this approximation, but I just do not get to the solution. Can anyone help me on the way?
     
  2. jcsd
  3. May 4, 2016 #2
    It is straightforward. Please let me know if you have not figured this out yet, since your post is quite old.
    Abe
     
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