Answer to Question #288503 in Calculus for Skshakeel

Question

Faced with two distinct demand functions:

Q1=24-0.2P1

Q2=10-0.05P2

Answer 1

Faced with two distinct demand functions:

$$Q_1=24-0.2P_1$$

$$Q_2=10-0.05P_2$$

The average cost $AC=40+35/Q.$ What price the firm will change with discrimination?

$$Q_1=24-0.2P_1$$ $$P_1=\frac{24-Q_1}{0.2}$$

$$Q_2=10-0.05P_2$$ $$P_2=\frac{10-Q_2}{0.05}$$

Average cost 

$$AC=40+\frac{35}{Q}$$

Total Revenue 

$$TR=P\times Q$$

$$TR_1=P_1\times Q_1=\frac{24-Q_1}{0.2}\times Q_1$$

$$=120Q_1-5Q_1^2$$

$$TR_2=P_2\times Q_2=\frac{10-Q_2}{0.05}\times Q_2$$

$$=200Q_2-20Q_2^2$$

Marginal revenue

$$MR_1=\frac{d(TR_1)}{dQ_1}=120-10Q_1$$

$$MR_2=\frac{d(TR_2)}{dQ_2}=200-40Q_2$$

Total cost

$$TC=(40-\frac{35}{Q})\times Q=40Q-35$$

Marginal cost

$$MC=\frac{d(TC)}{dQ}=40$$

Profit maximizing quantity of the firm will be when

$$MR_1=MR_2=MC$$

$$MR_1=MC=>120-10Q_1=40$$

$$Q_1=8$$

Then

$$P_1=\frac{24-Q_1}{0.2}=\frac{24-8}{0.2}=80$$

$$MR_2=MC=>200-40Q_2=40$$

$$Q_2=4$$

Then

$$P_2=\frac{10-Q_2}{0.05}=\frac{10-4}{0.05}=120$$

Therefore the prices firm will charge with discrimination are 

$$P_1=80\space and\space P_2=120$$