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 that the absolute value of the price elasticity of demand equals 3.
Here’s why:
In a monopoly, the firm maximizes profit by setting the price where marginal revenue (MR) equals marginal cost (MC). However, the price charged by the monopoly is usually higher than its marginal cost, and the monopoly’s ability to charge a price above marginal cost depends on the price elasticity of demand.
The relationship between price, marginal revenue, and elasticity of demand is described by the formula for marginal revenue:MR=P(1+1ε)MR = P \left( 1 + \frac{1}{\varepsilon} \right)MR=P(1+ε1)
Where:
- PPP is the price,
- ε\varepsilonε is the price elasticity of demand (in absolute terms),
- MRMRMR is marginal revenue.
For a monopolist, marginal revenue is also equal to marginal cost at the profit-maximizing level of output. So, the monopolist chooses its price by ensuring that:P=MC×(1+1ε)P = MC \times \left( 1 + \frac{1}{\varepsilon} \right)P=MC×(1+ε1)
Given that the monopolist charges a price that is three times its marginal cost (P=3×MCP = 3 \times MCP=3×MC), we can substitute into the equation:3×MC=MC(1+1ε)3 \times MC = MC \left( 1 + \frac{1}{\varepsilon} \right)3×MC=MC(1+ε1)
Simplifying:3=1+1ε3 = 1 + \frac{1}{\varepsilon}3=1+ε11ε=2\frac{1}{\varepsilon} = 2ε1=2ε=12\varepsilon = \frac{1}{2}ε=21
Thus, the absolute value of the price elasticity of demand is 3. This implies that the demand curve is elastic at the monopoly’s profit-maximizing price. When demand is elastic (i.e., ε>1\varepsilon > 1ε>1), a price increase leads to a proportionally larger decrease in quantity demanded, but because the monopolist is setting the price above marginal cost, the firm’s profits increase. In this case, the monopolist’s ability to price above marginal cost is a result of its market power and the elastic nature of the demand at the chosen price.
