
Jonathan T. answered 10/26/23
10+ Years of Experience from Hundreds of Colleges and Universities!
Let's address each part of the problem:
(a) To evaluate the demand elasticity E when p = 5, we'll use the formula for price elasticity of demand:
E = (dQ/dp) * (p/Q)
First, find the derivative of the demand function Q = 5000(6 - p) with respect to price p:
dQ/dp = d/dp [5000(6 - p)] = -5000
Now, plug this into the elasticity formula:
E(5) = (-5000) * (5 / (5000(6 - 5)))
E(5) = -5000 * (5 / 5000)
E(5) = -5
So, the demand elasticity when the price is $5 is E(5) = -5.
(b) To determine whether the price should be raised to increase revenue, we need to consider the elasticity of demand. In general, if demand is elastic (|E| > 1), increasing the price would lead to a decrease in total revenue. If demand is inelastic (|E| < 1), increasing the price would lead to an increase in total revenue.
In part (a), we found that the demand elasticity at p = 5 is E(5) = -5. Since |E| > 1, this means demand is elastic. Therefore, raising the price from $2 might lead to a decrease in total revenue.
(c) The demand elasticity is unitary when |E| = 1. To find the value of p for which the demand elasticity is unitary, set |E| equal to 1:
|E| = 1
|(-5000) * (p / (5000(6 - p)))| = 1
Now, solve for p:
5000p / (5000(6 - p)) = 1
Simplify the equation:
p / (6 - p) = 1
Cross-multiply:
p = 6 - p
2p = 6
p = 3
So, the demand elasticity is unitary when p = 3.
(d) To find the maximum revenue, you need to maximize the revenue function. The revenue R is given by:
R = p * Q
We already have the demand function Q = 5000(6 - p) and the price p = 3 from part (c) where elasticity is unitary.
R = 3 * Q
R = 3 * 5000(6 - 3)
R = 3 * 5000 * 3
R = 45,000