## Microeconomics Question from Walter E. Williams:

Given two isolated markets supplied by a single monopolist, let the two corresponding demand functions be:

${\displaystyle P_1=12-Q_1}$ and ${\displaystyle P_2=20-3Q_2}$.

The monopolist’s total cost function is:

${\displaystyle TC=3+2(Q_1+Q_2)}$.

(a) What will the prices be in each market?

(b) What will be the quantity sold in each market?

(c) What will be the total profits earned by the monopolist?

(a) Since we have two isolated markets, we can assume the monopolist will engage in price discrimination to maximize profits. As we are given both inverse demand functions, ${\displaystyle R_1=P_1Q_1=(12-Q_1)Q_1=12Q_1-Q_1^2}$, which implies ${\displaystyle MR_1=12-2Q_1}$. The same procedure gives ${\displaystyle MR_2=20-6Q_2}$. From the total cost function, we can derive ${\displaystyle MC=MC_1=MC_2=2}$.

By setting ${\displaystyle MR_1=MR_2=MC}$, we arrive at the solution for (b): ${\displaystyle 12-2Q_1=MR_1=MC=2}$, which implies ${\displaystyle Q_1=5}$; ${\displaystyle 20-6Q_2=MR_2=MC=2}$ implies ${\displaystyle Q_2=3}$.

With these market quantities, we can now determine respective market prices: ${\displaystyle P_1=12-5=7}$, ${\displaystyle P_2=20-3(3)=11}$.

(b) From (a) above: ${\displaystyle Q_1=5}$; ${\displaystyle Q_2=3}$.

(c) Profits will equal the aggregate revenues less total cost: ${\displaystyle \mathit{\Pi}^M=\sum_{i=1}^2 p_iq_i-TC=7(5)+11(3)-3-2(5+3)=35+33-19=49}$.

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