Chapter 5: Problem 3

Why is the demand curve with constant unitary elasticity concave?

Short Answer

Expert verified

The demand curve with constant unitary elasticity is concave because its second derivative with respect to price is positive. This indicates that the quantity demanded decreases at a slower rate when the price increases and increases at a slower rate when the price decreases. This corresponds with the constant unitary elasticity concept, where the percentage change in quantity demanded is equal to the percentage change in price.

Step by step solution

Define Price Elasticity of Demand

Price elasticity of demand (PED) is an economic measure that calculates the percentage change in the quantity demanded relative to the percentage change in price. It is used to determine how sensitive the quantity demanded is to price changes. Mathematically, PED can be calculated using the following formula: \(PED = \frac{ \% \text{ change in quantity demanded}}{\% \text{ change in price}} = \frac{\Delta Q / Q}{\Delta P / P}\) Where \(Q\) represents the quantity demanded, \(P\) represents the price, \(\Delta Q\) denotes the change in quantity demanded, and \(\Delta P\) denotes the change in price.

Describe Constant Unitary Elasticity

Constant unitary elasticity is a special case of price elasticity of demand where the percentage change in quantity demanded is the same as the percentage change in price. When PED = 1, we say that the demand is unit elastic. In other words, the quantity demanded changes proportionately with the price. For instance, if the price of a product increases by 10%, the quantity demanded will decrease by 10%. Conversely, if the price decreases by 10%, the quantity demanded will increase by 10%.

Analyze the Shape of the Demand Curve with Constant Unitary Elasticity

To prove that the demand curve with constant unitary elasticity is concave, we need to find the second derivative of the demand function with respect to price and show that it is positive. For a demand curve with constant unitary elasticity, the demand function can be represented as: \(Q = K \times P^{-1}\) Where \(K\) denotes a constant, \(Q\) represents the quantity demanded, and \(P\) represents the price.

Find the First Derivative of the Demand Function

To find the first derivative of the demand function, we will differentiate it with respect to the price: \(\frac{dQ}{dP} = K \times (-1) \times P^{-2}\) We can simplify this expression as follows: \(\frac{dQ}{dP} = -K \times P^{-2}\)

Find the Second Derivative of the Demand Function

Now, we will differentiate the first derivative with respect to the price to find the second derivative: \(\frac{d^2Q}{dP^2} = -K \times (-2) \times P^{-3}\) We can simplify this expression as follows: \(\frac{d^2Q}{dP^2} = 2K \times P^{-3}\)

Determine the Concavity of the Demand Curve with Constant Unitary Elasticity

Since the second derivative of the demand curve with respect to price, \(\frac{d^2Q}{dP^2} = 2K \times P^{-3}\), is positive (as both \(K\) and \(P^{-3}\) are positive), we can conclude that the demand curve with constant unitary elasticity is concave. The concavity of the demand curve in this case indicates that the quantity demanded decreases at a slower rate when the price increases, and increases at a slower rate when the price decreases. This matches the constant unitary elasticity concept, where the percentage change in quantity demanded is equal to the percentage change in price.

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