Question

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 (quantity sold) for the Ultra Mini is given by q1(p1, p2) = 100,000 ? 800p1 + 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: q2(p1, p2) = 150,000 + p1 ? 800p2
Find the prices for the Ultra Mini and the Big Stack that will maximize your total revenue.

Answer 1

Answer:

1. At p1 = (100,000 - p2)/1,600 for Ultra Minis

2. At p2 = (150,000 - p1)/1,600 for Big Stack

Step-by-step explanation:

Since we are dealing with demand functions in which there is a negative relationship between price and quantity demanded, the question marks marks in the two demand functions can be assumed to be negative signs. As a result, the equations can be re-stated as follows:

q1(p1, p2) = 100,000 - 800p1 + p2 ................................ (1)

q2(p1, p2) = 150,000 + p1 - 800p2 ............................... (2)

In economics, total revenue (TC) is quantity demanded/sold multiply by price, the TCs for Ultra Mini (TCq1), and the Big Stack (TCq2) can be obtained by multiplying equations (1) and (2) with p1 and p2 as follows:

For q1:

TCq1 = p1*q1(p1, p2) = p1(100,000 - 800p1 + p2)

TCq1 = 100,000p1 - 800p1^2 + p1p2 .............................. (3)

For q2:

TCq2 = p2*q2(p1, p2) = p2(150,000 + p1 - 800p2)

TCq2 = p2150,000 + p1p2 - 800p2^2 .......................... (4)

We will take partial derivatives of each of equations (3) and (4) to obtain the marginal revenue (MR) as follows:

Partial derivative of equation (3) with respect to p1 and equate to zero:

MR = dTCq1/dp1 = 100,000 - 2(800p1) + p2 = 0

                           = 100,000 - 1,600p1 + p2 = 0

By rearranging and solving for p1, we have:

1,600p1 = 100,000 - p2

p1 = (100,000 - p2)/1,600 ....................................... (5)

The p1 in equation (5) is the price that will maximize the total revenue of Ultra Mini.

Partial derivative of equation (4) with respect to p2 and equate to zero:

MR = dTCq2/dp2 = 150,000 + p1 - 2(800p2) = 0

                             = 150,000 - 1,600p2 + p1 = 0

By rearranging and solving for p2, we have:

1,600p2 = 150,000 - p1

p2 = (150,000 - p1)/1,600 ....................................... (6)

The p2 in equation (6) is the price that will maximize the total revenue of Big Stack.

Therefore the prices at which total revenue of the company will be maximized are at p1 = (100,000 - p2)/1,600 for Ultra Minis and at p2 = (150,000 - p1)/1,600 for Big Stack.