Answer to Question #283525 in Finance for Rajat

Question

The House of Music sells low cost turntables and speakers. The total revenue equation for sales the two products is given by

TR = QT – 6QT2 + 100 QS – 4QS2 + QTQS

where QT and QS are quantities of turntables and speakers, respectively. The marginal cost of turntables is $20 and the marginal cost of speakers is $10.

i) Are the two goods substitutes or complements?

ii) What is the profit-maximizing rate of output for each good?

iii) What would be the profit-maximizing rate of output if there were no demand interdependence between the two goods?

Answer 1

Solution:

i.). We use cross-price elasticity to determine whether goods are complements or substitutes; if the cross-price elasticity is positive, the goods are substitutes; if the cross-price elasticity is negative, the goods are complements.

The two products are complements since the cross-price elasticity is negative.

 

ii.). Profit maximizing output is where: MR = MC

MR₁ = $\frac{\partial TR} {\partial QT}$ = 1 - 12QT

MC₁ = 20

MR₁ = MC₁

1 - 12QT  = 20

 – 12QT = 19

QT = 19/-12 = -1.58

MR₂ = $\frac{\partial TR} {\partial QS}$ = 100 – 8QS

MC₂ = 10

MR₂ = MC₂

100 – 8QS = 20

100 – 20 = 8QS

80 = 8QS

QS = 10

iii.). Without demand interdependence:

TR = QT – 6QT2 + 100 QS – 4QS2 + QTQS

TR₁ = QT – 6QT2 + QT

MR₁ = $\frac{\partial TR_{1} } {\partial QT}$ = 1 - 12QT + 1

MC₁ = 20

MR₁ = MC₁

1 - 12QT + 1 = 20

12QT = 19

QT = $\frac{19}{12}$ = 1.58

 

TR₂ = 100QS – 4QS² + QS

MR₂ = $\frac{\partial TR_{2} } {\partial QS}$ = 100 - 8QS + 1

MC₂ = 10

MR₂ = MC₂

100 - 8QS + 1 = 20

101 – 20 = 8QS

81 = 8QS

QS = 10.1