Rotemberg Resource Constraint

Hello everyone,
I am a Phd student, not very familiar with DSGE frameworks.
My question is about the derivation ofthe aggregate resource constraint in the Rotemberg pricing setup.

I tried to derive it starting from the household budget constraint, but I am not really totally convinced this is the right procedure.

Thank you very much!

That should work. The price adjustment costs are bourne by firms, so they will enter the household budget constraint via firm dividends/profits.