# Profit-maximization for a monopoly

Between two countries, Richland and Poorland, with a strict ban on cross-border sales agreed with the Poorlandian government. The respective demand functions for both countries are:

$$Q_{poor} = 10 - P_{poor}$$ and $$Q_{rich} = 14 - 0.5P_{rich}.$$

The total cost, TC, in millions, depends on the total amount of units produced $$Q = Q_{rich} + Q_{poor}$$ and also includes 5 million of fixed costs.

TC $$= 5+2Q + \frac{Q^2}{8}$$

I. What is the profit-maximization strategy? Please indicate the prices and quantities, as well as the amount of social welfare generated.

II. Also are the prices above consistent with the rule of elasticities?

$$\textbf{My Idea:}$$ I took the derivatives of the TC to find MC. I isolated $$P$$ for poor and rich and multiplied it by $$Q$$ to find each revenue function, then subtracted it from TC to get the Total revenue, then I took the derivative to find MR = MC; which is this the maximization for a monopoly. Is this path the optimal one? Can someone please help me solve these? Thank you for your time and help.
A monopolist maximizes profit. For me, it is usually easier to do this in the quantity space. So you rearange the demand and maximize $$\max_{Q_p,Q_r} \quad P_p(Q_p)Q_p + P_r(Q_r)Q_r - TC(Q_p+Q_r)$$ $$\max_{Q_p,Q_r} \quad (Q_p-10)Q_p + (28-2Q_r)Q_r - 5-2(Q_r+Q_p) -\frac{(Q_r+Q_p)^2}{8}$$ The FOC gives you two equations with two unknowns, $$Q_r$$ and $$Q_p$$, which you solve for. Both FOC equations can be written in the form $$MR=MC$$. You then plug these $$Q$$s into the demand equations to get the prices. Next, you calculate consumer surplus and producer surplus and add them to get social welfare.