Answer to Question #169253 in Microeconomics for shreemanta kalita

Question

 A monopolist faces the demand curve Q = 120 - P/3. The cost function is C = Q 3 . Find the output that maximises this monopolist’s profits. What are the prices at profits and that output? Find the elasticity of demand at the profit maximising output.

Answer 1

Let's first write the inverse demand function:

$P=360-3Q.$

By the definition of the total revenue, we have:

$TR=PQ=(360-3Q)Q=360Q-3Q^2.$

Then, we can find the marginal revenue:

$MR=\dfrac{dTR}{dQ}=360-6Q.$

Let's find the marginal cost:

$MC=\dfrac{dTC}{dQ}=\dfrac{d}{dQ}(Q^3)=3Q^2.$

The monopolist maximises its profit when MC=MR:

$3Q^2=360-6Q,$ $Q^2+2Q-120=0.$

This quadratic equation has two roots: $Q_1=10$ and $Q_2=-12$. Since the quantity produced can't be negative the correct answer is $Q=10$.

Then, we can find the price that maximises the profit:

$P=360-3Q=360-3\cdot10=\$330.$

We can find the price elasticity of demand as follows:

$\epsilon=-\dfrac{P}{Q}\dfrac{dQ}{dP},$ $Q=\dfrac{P-360}{3}, \dfrac{dQ}{dP}=-\dfrac{1}{3},$ $\epsilon=-\dfrac{3P}{P-360}\cdot-(\dfrac{1}{3})=\dfrac{P}{P-360}.$

The price elasticity of demand when P = $330 equals:

$\epsilon=\dfrac{\$330}{\$330-360}=-11.$

Therefore, demand is elastic.