I am trying to run a fixed effects regression but unfortunately having some issues. I hope someone could help me.

I want to regress $P_{itc} = \mu_{c} + \phi_{b} + \gamma_{s} + \sigma_{1}*tax_{i,t} + CPI_{t-1} + \epsilon_{i,c,t}

I want to see the impact of tax (excise) on prices. The tax varies in quantities and in a small part by time (3 variations). The i is for product type (brand) and size, c is for city and t is for time (which is monthly). I want to include fixed effects by cities, by brand fixed effects and by differences in size.

1) My panel data is strongly unbalanced, meaning that for some cities I have more than one observation per product and for different years I have more observations than others. Any thoughts on this?

2) Given that I have CPI for the last period, does it make sense to include time fixed effects?

3) Originally I am tring to run: xtreg avg_price tax CPI_t-1 if quantity=600, fe cluster(id_city) to see the impact in different quantities. However, I am afraid that the variable tax has very small variation since it only changes in 3 points in time for 67 periods. Any thoughts on that?

Any answers would be highly appreciated. Many thanks.