A monopolist can produce any level of output at a cost of $\$10$

Question

In one market, the demand is $D_1=80-P_1$ and in the other $D_2=100-2P_2$. Suppose the markets are separated. What output level should be produced and what price will prevail. Since $D_1=80-P_1$, then $Profits=Revenue-Cost=Demand*Price-Demand*Cost=P_1(80-P_1)-10(80-P_1)$

For market 2, $D_2=100-2P_2$ so $Profit=P_2(100-2P_2)-10P_2$

To maximize profit just do calculus to work out the maximum of each of the resulting quadratics. Is this correct?

Answer 1

The inverse demand curve for Market 1 is given to be

$ P_1 = 80 -q_1$

Revenue function $= R_1(q_1) = p_1q_1 = (80-q_1)*q_1$

Marginal Revenue $MR_1 = 80-2q_1$

Similarly the inverse demand curve for Market 2 is given to be

$p_2 = 50-0.5q_2$

Revenue function $= R_2(q_2) = p_2q_2 = (50-0.5q_2)*q_2$

Marginal Revenue $MR_2 = p_2q_2 = (50 - q_2)$

Set $MR_1$ = Marginal Cost = 10

Set $MR_2$ = Marginal Cost = 10

At equilibrium, $80-2\hat q_1 = 10$

and

$50-\hat q_2 = 10$

Thus the output levels are $\hat q_1 = 35$ and $\hat q_2 = 40$

The equilibrium prices at the two markets are $\hat p_1 = 45$ and $\hat p_2 = 30$

Goodluck

Satish