Question

QUESTION

ABC Group manufactures smart phones that can be sold directly to retail outlets or to the Mother Company for further processing and eventual sale by them as a completely different model. The demand function for each of these markets is:

Retail Outlets: P₁ = 60 - 2Q₁

Mother Company: P₂ = 40 - Q₂

where P₁ and P₂ are the prices charged and Q₁ and Q₂ are the quantities sold in the respective markets.

ABC’s total cost function for the manufacture of this smart phone is:

TC = 10 + 8(Q₁ + Q₂)

1. Determine ABC’s total profit function.
2. What are the profit-maximising price and output levels for the product in the two markets?
3. At these levels of output, calculate the marginal revenue in each market.
4. What are ABC’s total profits if the firm is effectively able to charge different prices in the two markets?
5. Calculate the profit-maximising level of price and output if ABC is required to charge the same price per unit in each market. What are ABC’s profits under this condition?                                                                                                             

Answer 1

We have the following information

Demand equation (Retail Outlets): P₁ = 60 – 2Q₁

Demand equation (Mother Company): P₂ = 40 – Q₂

Total Cost (TC) = 10 + 8Q; where Q = Q₁ + Q₂

Marginal cost (MC) = ΔTC/ΔQ = 8

In this situation the equilibrium for the ABC Group will be the point where the marginal revenue from the two markets (MR₁ and MR₂) is equal to the marginal cost

MR₁ = MR₂ = MC

Total Revenue from Retail Outlets = Price × Quantity

TR₁ = P₁ × Q₁

TR₁ = (60 – 2Q₁)Q₁

TR₁ = 60Q₁ – 2Q²₁

MR₁ = ΔTR₁/ΔQ₁ = 60 – 4Q₁

Total Revenue from Mother Company: TR₂ = P₂ × Q₂

TR₂ = (40 – Q₂)Q₂

TR₂ = 40Q₂ – Q²₂

MR₂ = ΔTR₂/ΔQ₂ = 40 – 2Q₂

MR₁ = MC

60 – 4Q₁ = 8

Equilibrium quantity in the case of Retail Outlets: Q₁ = 13

MR₂ = MC

40 – 2Q₂ = 8

Equilibrium quantity in the case of Mother Company: Q₂ = 16

P₁ = 60 – 2Q₁

P₁ = 60 – 26

Equilibrium price in the case of Retail Outlets: P₁ = 34

P₂ = 40 – Q₂

P₂ = 40 – 16

Equilibrium price in the case of Mother Company: P₂ = 24

Total Profit = TR₁ + TR₂ – TC

Total Profit = 60Q₁ – 2Q²₁ + 40Q₂ – Q²₂ – 10 – 8(Q₁ + Q₂)

Total Profit = (60 × 13) – 2(13)² + (40 × 16) – (16)² – 10 – 8(13 + 16)

Total Profit = 780 – 338 + 640 – 256 – 10 – 232

Total Profit = $584

Now, it is given that the ABC Group charges same price. This means that ABC Group faces a single demand curve.

The single demand curve can be derived by horizontal summation of the two demand curve equations.

P₁ + P₂

60 – 2Q₁ + 40 – Q₂

P = 100 – 3Q the single demand curve being faced by ABC Group

P = Price

Q = Total Quantity (Q₁ + Q₂)

Total Revenue (TR) = Price × Quantity

TR = (100 – 3Q)Q

TR = 100Q – 3Q²

MR = ΔTR/ΔQ = 100 – 6Q

Equating the MR with the MC we get

100 – 6Q = 8

Equilibrium output = 15.3

P = 100 – 3Q

P = 100 – (3 × 15.3)

Equilibrium price = 54

Total Profit = TR – TC

Total Profit = (15.3 × 54) – 10 – (8 × 15.3)

Total Profit = 826.2 – 132.4

Total Profit = $693.8