So I'm trying to solve this problem: The country of Usland has 20,000 citizens. Recent events have polarized the country into two groups of equal...

So I'm trying to solve this problem:

The country of Usland has 20,000 citizens. Recent events have polarized the country into two groups of equal size: reds and blues.

Reds utility function is given by , where denotes the amount of good x that they individually consume, 

 denotes the total amount of good x consumed by all red citizens, and denotes the amount of money that they individually consume.

Blues utility function is given by , where  denotes the amount of good x that they individually consume, 

 denotes the total amount of good x consumed by all blue citizens, and  denotes the amount of money that they individually consume.

Good is produced by competitive firms at a constant marginal cost of $2/unit.

Question: What is the level of per-capita consumption of x for red citizens at the optimal allocation?

The government is considering introducing a Pigouvian subsidy system. Let  denote the subsidy in $/unit of consumption of good x for red citizens. Let  denote the subsidy in $/unit of consumption of good x for blue citizens.

Question: What is the value required for the market equilibrium to generate an optimal allocation? (In solving the problem,assume that individuals take aggregate consumption levels as given when making decisions)

I'm confused on the first part of the question, because it seems to me that the red and blue markets do not effect each other but they are buying from the same firm so they must be connected in some way. I'm not sure how to go about setting up equations such that both the red and blue markets are brought together in production with one firm. I believe I know how to find the Pigouvian tax once I solve the first part of the problem. Thank you!