Question

The following are two distinct demand functions: 𝑞1=24−0.2𝑃1 and 𝑞2=10−0.05𝑃2. The average cost AC=40+35/q. What price the firm will change with discrimination?

Answer 1

$Q_1=24-0.2P_1$

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

$Q_2=10-0.05P_2$

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

Average cost; $AC=40+\frac{35}{Q}$

Total Revenue; $TR=P\times Q$

so,

$TR_1=P_1\times Q_1$

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

$=120Q_1-5Q_1^2$

$TR_2=P_2\times Q_2$

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

$=200Q_2-20Q_2^2$

marginal revenue=First derivative of total revenue

$MR_1=\frac{dTR_1}{dQ_1}$

$=120-10Q_1$

$MR_2=\frac{dTR_2}{dQ_2}$

$=200-40Q_2$

Average cost; $AC=(40-\frac{35}{Q})\times Q$

$=40Q-35$

Marginal cost = First derivative of total cost

$MC=\frac{dTC}{dQ}$

$=40$

Profit maximizing quantity of the firm will be when;

$MR_!=MR_2=MC$

when;

$MR_1=MC\\$

$120-10Q_1=40\\10Q_1=120-40\\Q_1=8$

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

$=\frac{24-8}{0.2}$

$=80$

and when

$MR_2=MC\\200-40Q_2=40\\40Q_2=200-40\\Q_2=\frac{160}{40}\\Q_2=4$

then

$P_2=\frac{10-Q_1}{0.05}\\$

$=\frac{10-4}{0.05}$

$=120$

Therefore the prices firm will charge with discrimination are $P_1=80\space and\space P_2=120$