Question #283525

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?

Expert's answer

**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

MR_{1} = "\\frac{\\partial TR} {\\partial QT}" = 1 - 12QT

MC_{1} = 20

MR_{1} = MC_{1}

1 - 12QT = 20

– 12QT = 19

QT = 19/-12 = -1.58

MR_{2} = "\\frac{\\partial TR} {\\partial QS}" = 100 – 8QS

MC_{2} = 10

MR_{2} = MC_{2}

100 – 8QS = 20

100 – 20 = 8QS

80 = 8QS

QS = 10

iii.). Without demand interdependence:

TR = QT – 6QT2 + 100 QS – 4QS2 + QTQS

TR_{1} = QT – 6QT2 + QT

MR_{1} = "\\frac{\\partial TR_{1} } {\\partial QT}" = 1 - 12QT + 1

MC_{1} = 20

MR_{1} = MC_{1}

1 - 12QT + 1 = 20

12QT = 19

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

TR_{2} = 100QS – 4QS^{2} + QS

MR_{2} = "\\frac{\\partial TR_{2} } {\\partial QS}" = 100 - 8QS + 1

MC_{2} = 10

MR_{2} = MC_{2}

100 - 8QS + 1 = 20

101 – 20 = 8QS

81 = 8QS

QS = 10.1

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