<p>This simplification of the infinite sum is commonly made by a differencing approach. You can see <a href="https://economics.stackexchange.com/a/42258/27674">here</a> an example of this approach in this kind of models.</p>
<p>Regarding with the assumptions mentioned by you, for <span class="math-container">$\hat{mc}_t$</span> since it's a deviation from the levels variable steady state, in the equilibrium there's no expectations and therefore <span class="math-container">$mc_t=mc\implies \hat{mc}_t=0$</span> for all <span class="math-container">$t$</span> in the steady state. In other words in the steady state the value of this variable is deterministic and zero, but by no means the short term expectancy of this value <span class="math-container">$E_t[\hat{mc}_{t+k}]$</span> has to be zero, nor this is the way the series is simplified.</p>
<p>With respect to <span class="math-container">$\pi_t$</span>, something similar happens, since by definition <span class="math-container">$\pi_t=(1-\theta)(p_t^*-p_{t-1})$</span> (Galí, 2008, p.57) and again, in the steady state <span class="math-container">$p_t^*=p_{t-1}\implies \pi_t=0$</span>, but as with the marginal cost there's not a reason why log-linearized inflation is expected to be zero, apart from being in the steady state.</p>

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2021-02-07T22:12:54.817Z
2021-02-07T22:12:54.817Z
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<p>The below image is from P.47 of Monetary Policy, Inflation, and the Business Cycle by Jordi Gali. My question is &quot;how can we derive the equation (15). If (15) is a correct equation, my guess is that <span class="math-container">$E_t[\hat{mc}_{t+k}] = 0$</span> for all <span class="math-container">$k \not= 0$</span> and <span class="math-container">$E_t[\pi_{t+k}] = 0$</span> for all <span class="math-container">$k \not= 0,1$</span>. But, on what basis can we ensure these results?</p>
<p>Notation: <span class="math-container">$\theta \in (0,1)$</span> is the probability that the firm can change the price level. <span class="math-container">$\hat{mc}_{t+k} = mc_{t+k} - mc$</span> where <span class="math-container">$mc$</span> is the marginal cost at the steady state.</p>
<p>It is hard to put all relevant information to my question in this post. Here is the link for this book:<a href="https://perhuaman.files.wordpress.com/2014/06/gali_polc3adtica_monetaria.pdf" rel="nofollow noreferrer">https://perhuaman.files.wordpress.com/2014/06/gali_polc3adtica_monetaria.pdf</a> (please look at page 47)</p>
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<p><a href="https://i.sstatic.net/XCBWS.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/XCBWS.png" alt="enter image description here" /></a></p>
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<macroeconomics><new-keynesian-economics>
Jordi Gali Book First Edition Page 47 (need help for derivation)
shk910
# Jordi Gali Book First Edition Page 47 (need help for derivation)

(Asked by: username_0 on 2021-02-07)

The below image is from P.47 of Monetary Policy, Inflation, and the Business Cycle by Jordi Gali. My question is "how can we derive the equation (15). If (15) is a correct equation, my guess is that $E\_t[\hat{mc}\_{t+k}] = 0$ for all $k \not= 0$ and $E\_t[\pi\_{t+k}] = 0$ for all $k \not= 0,1$. But, on what basis can we ensure these results?

Notation: $\theta \in (0,1)$ is the probability that the firm can change the price level. $\hat{mc}\_{t+k} = mc\_{t+k} - mc$ where $mc$ is the marginal cost at the steady state.

It is hard to put all relevant information to my question in this post. Here is the link for this book:<https://perhuaman.files.wordpress.com/2014/06/gali_polc3adtica_monetaria.pdf> (please look at page 47)

> [![enter image description here](https://i.sstatic.net/XCBWS.png)](https://i.sstatic.net/XCBWS.png)

**Tags:** macroeconomics, new-keynesian-economics

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## 1 answers

### Answer by username_1 on 2021-02-07. Score: 1 | Accepted answer

This simplification of the infinite sum is commonly made by a differencing approach. You can see [here](https://economics.stackexchange.com/a/42258/27674) an example of this approach in this kind of models.

Regarding with the assumptions mentioned by you, for $\hat{mc}\_t$ since it's a deviation from the levels variable steady state, in the equilibrium there's no expectations and therefore $mc\_t=mc\implies \hat{mc}\_t=0$ for all $t$ in the steady state. In other words in the steady state the value of this variable is deterministic and zero, but by no means the short term expectancy of this value $E\_t[\hat{mc}\_{t+k}]$ has to be zero, nor this is the way the series is simplified.

With respect to $\pi\_t$, something similar happens, since by definition $\pi\_t=(1-\theta)(p\_t^\*-p\_{t-1})$ (Galí, 2008, p.57) and again, in the steady state $p\_t^\*=p\_{t-1}\implies \pi\_t=0$, but as with the marginal cost there's not a reason why log-linearized inflation is expected to be zero, apart from being in the steady state.


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