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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

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This simplification of the infinite sum is commonly made by a differencing approach. You can see here 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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