Your company manufactures two models of speakers the Ultra M

Your company manufactures two models of speakers, the Ultra Mini and the Big Stack. Demand for each depends partly on the price of the other. If one is expensive, then more people will buy the other. If p1 is the price of the Ultra Mini, and p2 is the price of the Big Stack, demand for the Ultra Mini is given by

q₁(p₁, p₂) = 100,000 400p₁ + p₂

where q₁ represents the number of Ultra Minis that will be sold in a year. The demand for the Big Stack is given by

₂(p₁, p₂) = 150,000 + p₁ 400p₂.

Find the prices for the Ultra Mini and the Big Stack that will maximize your total revenue. (Round your answers to two decimal places.)

p₁

p₂

Solution

demand for the Ultra Mini is given by

q1(p1, p2) = 100,000 400p1 + p2

The demand for the Big Stack is given by

q2(p1, p2) = 150,000 + p1 400p2.

TR of Ultra Mini:

= p1(100,000 400p1 + p2)

= 100000p1 - 400p1p1 + p1p2
= 100000p1 - 400p1^2 + p1p2

Partially differentiate with respect to p1
MR = 100000 - 800p1 + p2 .......Eq1

TR of Big Stack

= p2(150,000 + p1 400p2)

= 150000p2 + p1p2 - 400p2p2

Partially differentiate with respect to p2

MR = 150000 + p1 - 800p2 .......Eq2

Set Eq1 and Eq2 to zero (for max) and solve simultaneously:
100000 - 800p1 + p2 = 0
150000 + p1 - 800p2 = 0

p1 = 125.94 , p2 = 751.09