Question

Currently, the demand equation for necklaces is Q = 30 – 4P. The current price is $10 per necklace. Is this the best price to charge in order to maximize revenues? If not, what is?

Solve for the best price to charge in order to maximize revenues. Show any steps or processes used to reach the answer above. Explain your process as though you are teaching the concept to a student who is a beginner in economics.

Answer 1

The following steps should be used for the sum:

Step 1:

Price function should be arranged first. This is the rearranging of demand equation.

Q = 30 – 4P … demand equation

Hence,

30 – 4P = Q

-4P = Q – 30

4P = 30 – Q [by changing all signs]

P = 30/4 – Q/4

P = 7.5 – 0.25Q ….. Price function

Step 2:

Price function should be multiplied by Q in order to get total revenue (TR).

P = 7.5 – 0.25Q ….. Price function

TR = P × Q = 7.5Q – 0.25Q^2

TR = 7.5Q – 0.25Q^2

Step 3:

In case of maximizing revenues, derivative (calculus) of TR must be 0.

TR = 7.5Q – 0.25Q^2

Derivative of TR = [(power of Q × 7.5)Q^(1 – 1)] – [(power of Q × 0.25)Q^(2 – 1)]

0 = [(1 × 7.5)Q^(1 – 1)] – [(2 × 0.25)Q^(2 – 1)]

0 = (7.5Q^0) – (0.50Q)

0 = 7.5 × 1 – 0.50Q [since Q^0 = 1 as per indices rule]

0 = 7.5 – 0.50Q

0.50Q = 7.5

Q = 7.5/0.50 = 15

Step 4:

The value of Q (as found above) is to be placed in the price function in order to get the price.

P = 7.5 – 0.25Q ….. Price function

P = 7.5 – 0.25 × 15

   = 7.5 – 3.75

   = 3.75

Answer: the best price is $3.75 but not $10.