A faculty member sells Gatorade at UNC Charlotte football ga

A faculty member sells Gatorade at UNC Charlotte football games. He finds that if he raises his selling price by $1, he sells 5 less bottles each week. He turns over the operation to a Managerial Economics student for a week. The goal is to maximize revenue since the faculty member has a vast supply of Gatorade, and any unsold bottles can be sold in future weeks. The student takes over the operation for a week. She says she sold 30 bottles at the last game, and that is the optimal quantity that maximized revenue. What price did the student sell the Gatorade for?

Show your work/thought process:

Solution

Let the linear demand curve be:

P = a + bQ where P: Price ($), Q: Number of books

From data about Faculty member\'s actions:

P + 1 = a + b(Q - 5)

P + 1 = a + bQ - 5b

P = a + bQ - 5b - 1 .... (1)

Revenue = P x Q = aQ + bQ²

Revenue is maximized when dTR / dQ = 0

a + 2bQ = 0

a = - 2bQ

Since Q = 30 when TR is maximum,

a = - 60b

So, optimum price P = a + bQ = - 60b + b x 30 = - 60b + 30b = - 30b .... (2)

From (1),

P = a + bQ - 5b - 1

P = - 60b + 30b - 5b - 1

P = - 35b - 1

From (2),

- 30b = - 35b - 1

5b = - 1

b = - 1/5 = - 0.2

So, optimal price, P = - 30b = - 30 x (- 0.2) = 6