Calvo wage stickiness NK model optimal wage derivation

Greetings:

Following Costa (2016) in page 96, developing the labor variety optimal wage decided by the household, the FOC is:

0=E_t\sum_{i=0}^\infty(\beta\theta_w)^{t+i}\left\{ \begin{matrix} \psi_W\left[L_{t+i}\left(\frac{W_{t+i}}{W_{j,t}^\ast}\right)^{\psi_W}\right]^\varphi L_{t+i}\left(\frac{W_{t+i}}{W_{j,t}^\ast}\right)^{\psi_W}\left(\frac{1}{W_{j,t}^\ast}\right)\\+(1-\psi_W)\lambda_{t+i}L_{t+i}\left(\frac{W_{t+i}}{W_{j,t}^\ast}\right)^{\psi_W} \end{matrix} \right\}

I can reach this equation, nevertheless in the next equation from the book, the term \left(\frac{W_{t+i}}{W_{j,t}^\ast}\right)^{\psi_W}L_{t+i} is factored and cancelled out since the other side of the equation is 0. My question is whether this is valid, since W_{t+i} and L_{t+i} are functions of the infinite series since they’re indexed with i, then I would like to know if this is algebraically valid, or if it’s a mathematical typo. And if it’s valid, how can I derive the next equation, where the mentioned terms are cancelled out.

Thanks!

PD: If it helps:

-This is how L_{j,t+s} is defined: L_{j,t+s}=L_{t+s}\left(\frac{W_{t+s}}{W_{j,t}^\ast}\right)^{\psi_W}.

-W_{j,t}^\ast is not a function of the summation (check the indexes).

-The idea is to obtain a closed expression for W_{j,t}^\ast.

My first impression is that this looks like a typo. But I cannot help you, since I don’t have access to the book.

I also don’t see how you could cancel this term. See e.g. Appendix A of BP_2018_wage_pc.pdf - Google Drive

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This answer solved the issue, it seems that Costa (2016) assumes resulted cancellations of log-linearizations before actually log-linearizing. Your answer is very appreciated, thanks Dr. Pfeifer!

PD: If I may ask, is there a way to obtain closed form for W_{j,t}^\ast not log-linearizing? Seems to me like not because of the term V_{C,t+k|t}N_{t+k|t}\frac{(1-\tau^n_{t+k})}{(1+\tau^c_{t+k})}, but ask just to make sure.

You can express the sums recursively. See e.g. DSGE_mod/Derivation_Recursive_Pricing_Equation.pdf at master · JohannesPfeifer/DSGE_mod · GitHub

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Unfortunately, I made similar experiences with other papers.

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