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 the percentage change in quantity demanded is equal to the percentage change in price (E = -1), and the curve follows a power function of the form \(Q = kP^{-1}\). Taking the second derivative of this function, we find that it is positive (\(\frac{d^2Q}{dP^2} = 2kP^{-3}\)), which confirms the concave shape of the constant unitary elasticity demand curve.

Step by step solution

Define constant unitary elasticity

Constant unitary elasticity refers to a situation where the percentage change in quantity demanded is equal to the percentage change in price. In other words, the price elasticity of demand remains constant and equal to -1 for all points on the curve. Mathematically, price elasticity of demand is given by: \[E = \frac{\% \Delta Q}{\% \Delta P}\] For constant unitary elasticity, E = -1.

Define demand curve

The demand curve in economics represents the relationship between the quantity of goods demanded and their respective prices. The curve typically slopes downward, indicating that as the price of a good increases, the quantity demanded decreases, and vice versa.

Characteristics of constant unitary elasticity demand curve

A demand curve with constant unitary elasticity follows a power function of the form: \[Q = kP^{-1}\] where Q is the quantity demanded, P is the price, and k is a constant coefficient. If we plot this function on a price-quantity graph, we can observe a pattern that defines the concavity of the constant unitary elasticity demand curve.

Proving the concavity of the demand curve

To prove that a curve is concave, we can take the second derivative of the function and check if it's positive. First, we need to find the first derivative of the demand function: \[\frac{dQ}{dP} = -kP^{-2}\] Next, we'll take the second derivative of the demand function: \[\frac{d^2Q}{dP^2} = 2kP^{-3}\] Since the second derivative is positive, and the coefficient k is a constant with respect to price and quantity, the constant unitary elasticity demand curve is concave.

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