Following Costa, 2016 in page 96, developing the labor variety optimal wage decided by the household, the FOC is:
$$0=E_t\sum_{i=0}^\infty(\beta\theta_w)^{t+i}\left\{\psi_W\left[L_{t+i}\left(\frac{W_{t+i}}{W_{j,t}^\ast}\right)^{\psi_W}\right]^\varphi L_{t+i}\left(\frac{W_{t+i}}{W_{j,t}^\ast}\right)^{\psi_W}\left(\frac{1}{W_{j,t}^\ast}\right)+(1-\psi_W)\lambda_{t+i}L_{t+i}\left(\frac{W_{t+i}}{W_{j,t}^\ast}\right)^{\psi_W}\right\}$$
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 $W_{j,t}^\ast$.
