Assume that both the roommates A and B have to determine the amount they should spend on an AC. Obviously, the more money they put, better is the quality of AC.
Let C (G) denote the magnitude of public good measured by G.
Now, total money that they spent on an AC and their private consumption must be equal to the initial endowment.
Xa + Xb + C (G) = Wa + Wb
Being familiar with the words Pareto efficient, we now write this constraint as maximised utility of A such that utility of B remains unvarying.