Understanding Price Elasticity of Demand for a Rectangular Hyperbola

The question asks about the value of price elasticity of demand for a specific type of demand curve known as a rectangular hyperbola.

A demand curve is a graphical representation showing the relationship between the price of a good or service and the quantity demanded at that price. Price elasticity of demand measures how sensitive the quantity demanded is to a change in the price.

What is a Rectangular Hyperbola Demand Curve?

A rectangular hyperbola demand curve is characterized by the property that the total expenditure (or total revenue, Price × Quantity) remains constant at every point on the curve, regardless of the price change.

If P is the price and Q is the quantity demanded, then for a rectangular hyperbola demand curve:

\( P \times Q = k \)

where \( k \) is a constant value.

Price Elasticity of Demand Formula

The formula for price elasticity of demand (\( e \)) is typically given as:

\( e = - \frac{\text{Percentage change in quantity demanded}}{\text{Percentage change in price}} \)

Or, using calculus for point elasticity:

\( e = - \frac{dQ}{dP} \times \frac{P}{Q} \)

Calculating Elasticity for a Rectangular Hyperbola

For a rectangular hyperbola, we know \( P \times Q = k \), so \( Q = \frac{k}{P} \). To find \( \frac{dQ}{dP} \), we differentiate Q with respect to P:

\( \frac{dQ}{dP} = \frac{d}{dP} \left( \frac{k}{P} \right) = - \frac{k}{P^2} \)

Now, substitute this into the elasticity formula:

\( e = - \left( - \frac{k}{P^2} \right) \times \frac{P}{Q} \)

\( e = \frac{k}{P^2} \times \frac{P}{Q} \)

\( e = \frac{k}{P \times Q} \)

Since we know that for a rectangular hyperbola, \( P \times Q = k \), we can substitute this back into the elasticity equation:

\( e = \frac{k}{k} \)

\( e = 1 \)

This calculation shows that for any price and quantity combination on a rectangular hyperbola demand curve, the price elasticity of demand is always equal to 1. This means the demand is unit elastic at every point on the curve.

Interpreting the Result

When price elasticity of demand \( e = 1 \), it implies that a given percentage change in price leads to an equal percentage change in the quantity demanded. For example, if the price increases by 10%, the quantity demanded decreases by exactly 10%, keeping the total expenditure constant.

Let's summarize the options and the result:

Our derivation confirms that for a rectangular hyperbola demand curve, the price elasticity of demand is always equal to 1.