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.
