Question

Why is the demand curve with constant unitary elasticity concave?

Short answer

The demand curve with constant unitary elasticity is concave due to the relationship between price and quantity demanded represented in the equation \(Q = kP^{-1}\). Constant unitary price elasticity of demand means the percentage change in quantity demanded is always equal to the percentage change in price, and the negative sign shows their inverse relationship. By calculating the second derivative of the function, which is positive, we can confirm that the demand curve is concave.

Step-by-step solution

Understanding constant unitary elasticity

Constant unitary price elasticity of demand refers to a situation where the percentage change in quantity demanded is always equal to the percentage change in price. Mathematically, it can be expressed as: \(E_d = \frac{%\Delta Q}{%\Delta P} = -1\) Where \(E_d\) is the price elasticity of demand, %\Delta Q is the percentage change in quantity demanded, and %\Delta P is the percentage change in price. The negative sign shows the inverse relationship between price and quantity demanded, meaning as the price increases, the quantity demanded decreases, and vice versa.

Understanding concave curves

A concave curve, also known as a concave function, is a curve that bends inwards. In the context of a demand curve with constant unitary elasticity, the curve will be concave towards the origin of the graph. This means that as we move along the curve from left to right, the slope of the curve becomes less steep, i.e., the second derivative of the curve with respect to the independent variable (price) is negative.

Connecting constant unitary elasticity to concave curves

To show that the demand curve with constant unitary elasticity is concave, let's consider its equation: \(Q = kP^{-1}\) Where Q is the quantity demanded, P is the price, and k is a constant. Now, we will calculate the second derivative of this function with respect to the price (P). First, we'll calculate the first derivative: \(\frac{dQ}{dP} = -kP^{-2}\) Now, let's calculate the second derivative: \(\frac{d^2Q}{dP^2} = 2kP^{-3}\) Since the second derivative is positive for any value of P (price) and k (constant), the demand curve is always concave.

Conclusion

In conclusion, the demand curve with constant unitary elasticity is concave because of the relationship between price and quantity demanded represented in the equation \(Q = kP^{-1}\). By finding the second derivative, which is positive, it can be confirmed that the curve is concave.