Profit-Maximizing Level of Price and Output

Philips Industries manufactures a certain product that can be sold directly to retail outlets or to the Superior Company for further processing and eventual sale as a completely different product. The demand function for each of these markets is

Retail Outlets: P1 = 60 -2Q1

Superior Company: P2 = 40 -Q2

where P1 and P2 are the prices charged and Q1 and Q2 are the quantities sold in the respective markets. Phillip's total cost function for the manufacture of this product is

TC = 10 + 8(Q1 + Q2)

a. Determine Phillip's total profit functions.
b. What are the profit-maximizing price and output levels for the product in the two markets?
c. At these levels of output, calculate the marginal revenue in each market.
d. What are Phillips's total profits if the firm is effectively able to charge different prices in the two markets?
e. Calculate the profit-maximizing level of price and output if Phillips is required to charge the same price per unit in each market. What are Phillips's profits under this condition?
a.

Total Revenue from Retail outlets=Price*Quantity=(60-2Q1)*Q1
TR1=60Q1-2Q1^2

Total Revenue from Superior Company=Price*Quantity=(40-Q2)*Q2
TR2=40Q2-Q2^2

Total Cost, TC=10+8(Q1+Q2)

Profit=Total Revenue-Total Cost
=TR1+TR2-TC
=60Q1-2Q1^2+40Q2-Q2^2-[10+8(Q1+Q2)]
=60Q1-2Q1^2+40Q2-Q2^2-10-8Q1-8Q2
=52Q1+32Q2-2Q1^2-Q2^2-10

b.

Marginal Cost for Market 1(Retail outlets)=MC1=d(TC)/dQ1=8
TR1=60Q1-2Q1^2
Marginal Revenue for Market 1=MR1=d(TR1)/dQ1=60-4Q1
For Profit maximization, put MR1=MC1
60-4Q1=8
4Q1=52
Q1=13
P1=60-2Q1=60-2*13=34

Marginal Cost for Market 2(Superior Company)=MC2=d(TC)/dQ2=8
TR2=40Q2-Q2^2
Marginal Revenue for Market 2=d(TR2)/dQ2=40-2Q2
For Profit maximization, put MR2=MC2
40-2Q2=8
2Q2=32
Q2=16
P2= 40-Q2=40-16=24

For Retail Outlets, Profit maximization output is Q1=13 and corresponding price P1 is 34.
For Superior Company, Profit maximization output is Q2=16 and corresponding price P2 is 24.

c.
MR1 (Retail Outlets)=60-4Q1=60-4*13=8
MR2 (Superior Company)=40-2Q2 =40-2*16=8