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 theyâ€™re indexed with i, then I would like to know if this is algebraically valid, or if itâ€™s a mathematical typo. And if itâ€™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.