If a profit-maximizing monopoly charges a price that is three times its marginal cost, the demand function at the profit-maximizing quantity must be elastic because the absolute value of the price elasticity of demand equals

If a profit-maximizing monopoly charges a price that is three times its marginal cost, the demand function at the profit-maximizing quantity must be elastic because the absolute value of the price elasticity of demand equals

The Correct Answer and Explanation is:

The correct answer is: Elastic.

Explanation:

In the context of monopoly pricing, the monopolist maximizes profits by choosing a quantity where marginal revenue equals marginal cost (MR = MC) and then sets the price corresponding to this quantity on the demand curve. The relationship between price (P), marginal cost (MC), and the price elasticity of demand (E) is crucial for understanding the monopolist’s pricing behavior.

The monopolist’s pricing rule can be derived from the formula for marginal revenue (MR), which is related to the price elasticity of demand. The formula for marginal revenue in terms of elasticity of demand is:MR=P(1−1∣E∣)MR = P \left(1 – \frac{1}{|E|}\right)MR=P(1−∣E∣1​)

where:

• PPP is the price charged by the monopolist,
• ∣E∣|E|∣E∣ is the absolute value of the price elasticity of demand, and
• MRMRMR is marginal revenue.

When a monopoly charges a price that is three times its marginal cost, we have:P=3×MCP = 3 \times MCP=3×MC

Substituting this into the marginal revenue equation:MR=3×MC(1−1∣E∣)MR = 3 \times MC \left( 1 – \frac{1}{|E|} \right)MR=3×MC(1−∣E∣1​)

At the profit-maximizing quantity, the monopolist sets MR=MCMR = MCMR=MC. Thus, substituting MR=MCMR = MCMR=MC into the equation, we get:MC=3×MC(1−1∣E∣)MC = 3 \times MC \left( 1 – \frac{1}{|E|} \right)MC=3×MC(1−∣E∣1​)

Dividing both sides by MCMCMC (assuming MC≠0MC \neq 0MC=0):1=3(1−1∣E∣)1 = 3 \left( 1 – \frac{1}{|E|} \right)1=3(1−∣E∣1​)

Simplifying:1=3−3∣E∣1 = 3 – \frac{3}{|E|}1=3−∣E∣3​3∣E∣=2\frac{3}{|E|} = 2∣E∣3​=2∣E∣=32|E| = \frac{3}{2}∣E∣=23​

Since the price elasticity of demand is ∣E∣=1.5|E| = 1.5∣E∣=1.5, which is greater than 1, this indicates that the demand at the monopolist’s profit-maximizing price is elastic. This means that a 1% increase in price will lead to a greater than 1% decrease in quantity demanded.

In summary, the monopolist charges a price that is above marginal cost, and for this to be consistent with profit-maximizing behavior, the demand curve at the chosen price must be elastic. This is because the monopolist will only choose a price above marginal cost when the price elasticity of demand is elastic, ensuring that the revenue loss from lower sales is outweighed by the higher price.