True/False: A demand curve with constant slope over all quantity values can have a continuously changing price elasticity of demand
The correct answer and explanation is:
Answer: True
A demand curve with a constant slope can indeed have a continuously changing price elasticity of demand.
Explanation (300 words):
The price elasticity of demand (PED) measures how much the quantity demanded of a good responds to a change in its price. Mathematically, it is defined as: PED=∣dQdP⋅PQ∣\text{PED} = \left| \frac{dQ}{dP} \cdot \frac{P}{Q} \right|
- dQdP\frac{dQ}{dP}: the derivative of quantity with respect to price (i.e., the slope of the demand curve).
- PP: price
- QQ: quantity demanded
Now consider a linear demand curve, such as: Q=a−bPQ = a – bP
This equation represents a straight line with a constant slope of −b-b (since dQ/dP=−bdQ/dP = -b).
However, the price elasticity of demand at any point on this demand curve is: PED=∣−b⋅PQ∣=∣−b⋅Pa−bP∣\text{PED} = \left| \frac{-b \cdot P}{Q} \right| = \left| \frac{-b \cdot P}{a – bP} \right|
This shows that although the slope −b-b is constant, the elasticity changes continuously as you move along the curve because both PP and QQ are changing.
- At high prices (low quantities), P/QP/Q is large, so elasticity is high.
- At low prices (high quantities), P/QP/Q is small, so elasticity is low.
At the midpoint of the linear demand curve, the price elasticity is exactly 1 (unit elastic). Above the midpoint, demand is elastic (PED > 1), and below it, demand is inelastic (PED < 1).
Conclusion:
Even though the slope of a linear demand curve is constant, the price elasticity of demand varies continuously along the curve because it depends not only on the slope but also on the ratio of price to quantity at each point. Therefore, the statement is True.