Answer and Explanation:
To calculate the point price elasticity of demand at the profit maximizing price ($25) and quantity (5,090) using the estimated demand function Q = 8,398 - 247P, we can follow these steps:
Step 1: Calculate the quantity demanded at the profit maximizing price ($25):
Q = 8,398 - 247P
Q = 8,398 - 247(25)
Q = 8,398 - 6,175
Q ≈ 2,223
Step 2: Calculate the slope of the demand function at the profit maximizing price:
The slope of the demand function is the coefficient of P, which is -247.
Step 3: Calculate the price elasticity of demand using the formula:
E = (ΔQ / Q) / (ΔP / P)
Since we're calculating the point price elasticity of demand, we can substitute ΔQ with 0 (since we're evaluating elasticity at a specific point) and ΔP with 1 (a small change in price).
E = (0 / 2,223) / (1 / 25)
E = 0 / 0.04
E = 0
a. Calculate the markup at the profit maximizing price:
Markup = Profit Maximizing Price - Cost Price
Markup = $25 - Cost Price
b. Compare the markup and elasticity from parts a and b:
To determine if the price calculated in part b is the profit maximizing price, we need to compare the markup and elasticity.
If the elasticity is less than 1, it suggests that the demand is inelastic and a higher markup can be applied. If the elasticity is greater than 1, it suggests that the demand is elastic and a lower markup should be applied.
Since the price elasticity of demand is 0, it indicates that the demand is perfectly inelastic. This means that the price can be increased without impacting the quantity demanded.