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:
In the context of a profit-maximizing monopoly, if the price is three times the marginal cost, this implies that the monopoly is operating in the elastic range of the demand curve.
To understand why, let’s first define the relevant economic concepts:
- Price elasticity of demand (PED) measures the responsiveness of the quantity demanded to a change in price. If the absolute value of the price elasticity of demand (|PED|) is greater than 1, the demand is elastic. If it’s less than 1, demand is inelastic. If it equals 1, the demand is unit elastic.
- Monopoly pricing: A monopoly maximizes its profit by setting the quantity where marginal cost (MC) equals marginal revenue (MR). The monopoly then charges the price based on the demand curve at that quantity. The price charged is typically higher than the marginal cost because the monopoly faces a downward-sloping demand curve, meaning it must lower the price on all units sold to increase sales.
Now, when a monopoly charges a price that is three times its marginal cost, this relationship indicates that the monopoly is operating in the elastic portion of the demand curve. To derive this:
- Markup Rule: The monopoly’s markup over marginal cost is given by the Lerner Index, which is calculated as: Lerner Index=P−MCP\text{Lerner Index} = \frac{P – MC}{P}Lerner Index=PP−MC Where:
- P is the price charged
- MC is the marginal cost
- The Lerner Index is also related to the price elasticity of demand through the formula: P−MCP=1∣PED∣\frac{P – MC}{P} = \frac{1}{|PED|}PP−MC=∣PED∣1 By equating the two expressions for the Lerner Index, we have: 23=1∣PED∣\frac{2}{3} = \frac{1}{|PED|}32=∣PED∣1 Solving for |PED| gives: ∣PED∣=32=1.5|PED| = \frac{3}{2} = 1.5∣PED∣=23=1.5
Since the absolute value of the price elasticity of demand (|PED|) is 1.5, which is greater than 1, the demand curve is elastic at the profit-maximizing price and quantity.
Conclusion:
The price elasticity of demand at the monopoly’s profit-maximizing output is greater than 1, meaning the demand is elastic. This is why the monopoly can set a price that is three times the marginal cost while still maximizing its profit.
