Question

A monopolist’s demand function is given as Q,=2000-10P where Q is the quantity is produced and sold and P is the price per unit in Ksh. If the firm’s marginal cost is K.sh100:

1. Calculate the monopolist’s equilibrium quantity and price.   

Answer 1

Monopolist demand function is

Q=2000 -10P

Inverse demand function is

$10P= 2000 -Q \\ P= 200 - \frac{Q}{10}$

Total revenue= Price $\times$ Quantity

$= (200 - \frac{Q}{10})Q \\ = 200Q -\frac{Q^2}{10}$

$Marginal revenue = \frac{\partial TR}{\partial Q} \\ MR= 200 - 2 \times \frac{Q}{10} \\ = 200 - \frac{Q}{5}$

MC= 100 given

Monopolist equilibrium condition

$MR= MC \\ 200 - \frac{Q}{5}=100 \\ \frac{Q}{5} = 200 -100 \\ \frac{Q}{5}= 100 \\ Q = 100 \times 5 \\ Q = 500$

Price equation

$P= 200 - \frac{Q}{10}$

Substituting Q into price equation

$P= 200 - \frac{500}{10} \\ P= 200 -50 \\ P= 150$

Equilibrium price will be 150 and quantity will be 500 units.