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## Microeconomics Question from Walter E. Williams:Edit

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

$P_1=12-Q_1$ and $P_2=20-3Q_2$.

The monopolist’s total cost function is:

$TC=3+2(Q_1+Q_2)$.

(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, $R_1=P_1Q_1=(12-Q_1)Q_1=12Q_1-Q_1^2$, which implies $MR_1=12-2Q_1$. The same procedure gives $MR_2=20-6Q_2$. From the total cost function, we can derive $MC=MC_1=MC_2=2$.

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

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

(b) From (a) above: $Q_1=5$; $Q_2=3$.

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

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