The Andrews Corporation produced satellite dishes, was purchased by another compay, which was a cable company: The Total Cost function for the New Company is: TC = 1,000 + 16Q – 16Q2, where the demand curve for Satellite Dishes is PS = 400 – QS and that for the cable component of the firm is PC = 300 – 3QC.
1. Determine the production requirements for each product of the multi-product firm, which should be produced each period.
2. What price should be charged for each product?
To determine the production requirements for each product of the multi-product firm, we need to find the quantities that maximize the firm's total profit. This can be done by maximizing the firm's profit function, which is calculated by subtracting the total cost (TC) from the total revenue (TR).
1. Production Requirements:
Step 1: Calculate the total revenue function (TR):
Total revenue (TR) equals the product of the price and quantity for each product.
TR = PS * QS + PC * QC
Step 2: Calculate the profit function (π):
Profit (π) is equal to total revenue minus total cost.
π = TR - TC
Step 3: Determine the production requirements:
To find the production requirements, we need to maximize the profit function by taking its derivative with respect to each product's quantity and setting it equal to zero.
Take the derivative of the profit function with respect to each product's quantity and set them equal to zero:
dπ/dQS = d(TR - TC)/dQS = 0
dπ/dQC = d(TR - TC)/dQC = 0
Solve the two equations for QS and QC to get the production requirements.
2. Price for Each Product:
To find the price for each product, we need to substitute the production requirements back into their respective demand curves and solve for price.
For the satellite dishes (PS):
Given demand curve: PS = 400 - QS
Substitute the calculated value of QS into the demand curve equation to find the price.
For the cable component (PC):
Given demand curve: PC = 300 - 3QC
Substitute the calculated value of QC into the demand curve equation to find the price.
By following these steps, you will be able to determine the production requirements for each product and the corresponding price for each product.