Profit-maximization for a monopoly

Question

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?

Can someone please help me with this?

$\textbf{My Idea:}$ 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.

Answer 1

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.