Answer:
1. For Ultra Minis, the equation is p1 = (100,000 - p2)/1,600
2. For Big Stack, the equation is p2 = (150,000 - p1)/1,600
Step-by-step explanation:
Given the demand equations which reflect an inverse correlation between price and the quantity demanded, the placeholders in both equations should be interpreted as negative signs. So, we can rewrite the functions as:
q1(p1, p2) = 100,000 - 800p1 + p2................................ (1)
q2(p1, p2) = 150,000 + p1 - 800p2............................... (2)
Total revenue (TR) in economics is defined as the quantity sold multiplied by the price, hence we can calculate the total revenues for Ultra Mini (TRq1) and Big Stack (TRq2) by multiplying equations (1) and (2) by p1 and p2 respectively:
For q1:
TRq1 = p1 * q1(p1, p2) = p1(100,000 - 800p1 + p2)
TRq1 = 100,000p1 - 800p1^2 + p1p2.......................... (3)
For q2:
TRq2 = p2 * q2(p1, p2) = p2(150,000 + p1 - 800p2)
TRq2 = p2150,000 + p1p2 - 800p2^2........................ (4)
To find marginal revenues (MR), we will differentiate each of these revenue functions:
For equation (3), taking the partial derivative with respect to p1 and setting it to zero:
MR = dTRq1/dp1 = 100,000 - 2(800p1) + p2 = 0
= 100,000 - 1,600p1 + p2 = 0
Rearranging gives:
1,600p1 = 100,000 - p2
p1 = (100,000 - p2)/1,600.................................. (5)
In equation (5), p1 represents the price that maximizes Ultra Mini's total revenue.
Now differentiating equation (4) with respect to p2 and setting it to zero:
MR = dTRq2/dp2 = 150,000 + p1 - 2(800p2) = 0
= 150,000 - 1,600p2 + p1 = 0
Rearranging yields:
1,600p2 = 150,000 - p1
p2 = (150,000 - p1)/1,600.................................... (6)
The value for p2 in equation (6) is the price that optimizes the total revenue for Big Stack.
Thus, the optimal prices for maximizing total revenue are p1 = (100,000 - p2)/1,600 for Ultra Minis and p2 = (150,000 - p1)/1,600 for Big Stack.
