Question

A price-discriminating monopolist faces the following inverse demand functions: In Market One it is P1- 80-Q1 and in Market Two it is P2 60-Q2 Marginal cost is constant at $10. Consumers in market two can resell the good to consumers in market one at a cost of $4 per unit. Find the profit-maximizing quantity and price charged in each market subject to the resale constraint.please help solve. Is this also 3rd degree price discrimination?

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Maximize profits given the resale constraint where the price in market 1 exceeds the price in market 2 by $4 or the resale cost or P1 = P2 + 4

Maximize Π = P1Q1 + P2Q2 – MC1Q1 – MC1Q2 + λ(P1 – P2 – 4)

Π = (80 – Q1)Q1 + (60 – Q2)Q2 – 10Q1 – 10Q2 + λ(80 – Q1 – 60 + Q2 – 4)

Π’(Q1) = 0 or 80 – 2Q1 – 10 – λ or λ = 70 – 2Q1

Π’(Q2) = 60 – 2Q2 – 10 + λ or λ = 2Q2 – 50

Q1 – Q2 = 16

Solve these equations

70 + 50 = 2Q2 + 2Q2 and Q1 – Q2 = 16

Q1 + Q2 = 60… (1) and Q1 – Q2 = 16… (2)

16 + Q2 + Q2 = 60 or 2Q2 = 44.

This gives Q2 = 22 units and Q1 = 38 units. Prices are P1 = $42 and P2 = $38.

Maximize profits given the resale constraint where the price in market 1 exceeds the price in market 2 by $4 or the resale cost or P1 P2 4 Π (80-QIIQ1+ (60-Q2Q2-10Q1-10Q2+ λ(80-Q1-60 + Q2-4) 「「(Q2) = 60-2Q2-10 + λ or λ=2Q2-50 Q1- Q2 16 Solve these equations 70 + 50 = 2Q2 + 2Q2 and Q1-Q2 = 16 Q1+ Q2 = 60 (1) and Q1-Q2 = 16 (2) 16 + Q2+ Q2 = 60 or 2Q2-44. This gives Q2 22 units and Q1 38 units. Prices are P1 - S44 and P2 $38

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