Question

A monopolist charges different prices in the two markets where his demand functions are x1 = 21-0.1 p1 and x2 = 50 - 0.4 p2 , p1 and p2 being prices and 1 and x2

quantities demanded. His total cost function is TC = 10x + 2000, where x is total output. Find the prices that the monopolist should charge to maximize his profit. Also, verify

that higher price will be charged in the market having the lower price elasticity of demand by calculating the price elasticity of demand using the prices and the respective

demand functions.

Answer 1

Maximum profit prices:

Revenue (R):

$R_1=P_1x_1=p_1(21-0.1p_1)=21p_1-0.1p_1^2$

$R_2=P_2x_2=p_2(50-0.4p_2)=50p_2-0.4p_2^2$

$TC=10x+2000=10x_1+10x_2+2000=10(21-0.1p_1)+10(50-0.4p_2)+2000=2710-p_1-4p_2$

$Profit=TR-TC=R_1+R_2-TC=21p_1-0.1p_1^2+50p_2-0.4p_2^2-2710+p_1+4p_2=22p_1+54p_2-0.1p_1^2-0.4p_2^2-2710$

$\frac{\delta p}{\delta p_1}=22-0.2p_1=0$ $p_1=110$

$\frac{\delta p}{\delta p_1}=54-0.8p_2=0$ $p_2=67.5$

verification:

$\frac{\delta^2 p}{\delta p_1^2}=-0.2$

$\frac{\delta^2 p}{\delta p_2^2}=-0.8$

$\frac{\delta^2 p}{\delta p_1\delta p_2}=0$

$-0.2*-0.8-0^2=0.16>0$

The function will have a maximum value if $p_1=110\ and\ p_2=67.5$