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.