Question

For each demand function, find an expression for the price elasticity of demand. The answer will typically be a function of the price, p. As an example, consider the linear demand curve, D(p) = 30 − 6p. Then dD(p)/dp = −6 and p/q = p/(30 − 6p), so the price elasticity of demand is −6p/(30 − 6p).

a. D(p) = 60 − p. dD(p)/dp = −1
b. D(p) = a − bp. dD(p)/dp = −b
c. D(p) = 40p−2. dD(p)/dp = 80p-3
d. D(p) = Ap−b. dD(p)/dp = -bAp-b-1
e. D(p) = (p + 3)−2 dD(p)/dp = -2(p+3)
f. D(p) = (p + a)−b. dD(p)/dp = −b(p + a)-b-1

Answer 1

The price elasticity of demand (PED) for each demand function is obtained using the formula (dD(p)/dp) × (p / D(p)). The PED expressions for the given functions represent the responsiveness of quantity demanded to price changes.

Step-by-step explanation:

To find the expression for the price elasticity of demand for each demand function, we need to use the formula for price elasticity of demand (PED), which is the percentage change in quantity demanded divided by the percentage change in price. When applied to a demand function, this translates to the formula:

(dD(p)/dp) × (p / D(p)).

Let's calculate the PED for each of the given demand functions:

• a. For the demand function D(p) = 60 - p, with dD(p)/dp = -1, the PED is:
• PED = (-1) × (p / (60 - p)).
• b. For the demand function D(p) = a - bp, with dD(p)/dp = -b, the PED is:
• PED = (-b) × (p / (a - bp)).
• c. For the demand function D(p) = 40p-2, with dD(p)/dp = -80p-3, the PED is:
• PED = (-80p-3) × (p / (40p-2)).
• d. For the demand function D(p) = Ap-b, with dD(p)/dp = -bAp-b-1, the PED is:
• PED = (-bAp-b-1) × (p / (Ap-b)).
• e. For the demand function D(p) = (p + 3)-2, with dD(p)/dp = -2(p+3)-3, the PED is:
• PED = (-2(p + 3)-3) × (p / ((p + 3)-2)).
• f. For the demand function D(p) = (p + a)-b, with dD(p)/dp = -b(p + a)-b-1, the PED is:
• PED = (-b(p + a)-b-1) × (p / ((p + a)-b)).

These expressions represent how the quantity demanded changes in response to a change in price.