Q.A monopolist selling in two separate markets, A and B, has a constant Marginal Cost of Rs. 10. The demand functions are $P_{A}=50-Q_{A}$ in Market A and $P_{B}=40-2Q_{B}$ in Market B. Find the profit-maximising price and quantity in each market, verify that $MR_A=MR_B=MC$, and calculate total profit.

Find MR in each market separately, set each equal to the common MC to find each market's quantity and price, then verify the two MR values are equal and sum the profits.

Market A: $TR_A=50Q_A-Q_A^2$, $MR_A=50-2Q_A$. Set $=10$: $Q_A=20$, $P_A=30$.

Market B: $TR_B=40Q_B-2Q_B^2$, $MR_B=40-4Q_B$. Set $=10$: $Q_B=7.5$, $P_B=25$. …

Market A: $TR_A=P_A\times Q_A=(50-Q_A)Q_A=50Q_A-Q_A^{2}$, so $MR_A=50-2Q_A$. Setting $MR_A=MC=10$:

Market B: $TR_B=P_B\times Q_B=(40-2Q_B)Q_B=40Q_B-2Q_B^{2}$, so $MR_B=40-4Q_B$. Setting $MR_B=MC=10$:

Verification of the price-discrimination rule: $MR_A=50-2(20)=10$ and $MR_B=40-4(7.5)=10$ — both equal Rs. 10, which is also equal to MC, exactly satisfying $MR_A=MR_B=MC$.

Total profit:

$\text{Profit}=(P_A-MC)Q_A+(P_B-MC)Q_B=(30-10)(20)+(25-10)(7.5)=400+112.5=Rs.\,512.5$ …

As a cross-check, if the same 27.5 total units were instead sold as a single undiscriminated batch split evenly and arbitrarily (say Q_A=15, Q_B=12.5) instead of at the MR-equalising split, the two markets' marginal revenues would NOT be equal (MR_A at Q_A=15 is 50-30=20, MR_B at Q_B=12.5 is 40-50=-10) — since MR_A > MR_B here, shifting units from B to A …

A common mistake is charging the SAME price in both markets (treating them as one combined market) — this fails to exploit the different elasticities of demand and yield …