Question

A firm produces two different goods, with demand given by the following: Pa = 200 – 25Qa – 2Qb and Pb = 75 – 2Qb Where Pa = price of good A, Pb = price of good B, Qa = quantity of good A and Qb = quantity of good B.

The marginal costs for the two goods are 12 for good A and 20 for good B.

Determine optimal prices and quantities for each good.

Answer 1

Inverse demand for good A: Pa = 200 – 25Qa – 2Qb

Inverse demand for good B: Pb = 75 – 2Qb

Total revenue from good A (TRa)=(200 – 25Qa – 2Qb)*Qa=200Qa-25Qa^2-2QaQb

Marginal revenue from good A (MRa)= d(TRa)/dQa=200-50Qa-2Qb

Total revenue from good B (TRb)=(75 – 2Qb)*Qb=75Qb-2Qb^2

Marginal revenue from good B (MRb)=d(TRb)/dQb=75-4Qb

For optimization the frim will equate the marginal revenue for each good to their respective marginal cost

MRb=MCb: 75-4Qb=20. SOlving for Qb, we Qb=13.75

Substituting this Qb=13.75 in the MRa and then equating MRa=MCa: 200-50Qa-2*13.75=12. Thus solving for Qa we get Qa=3.21

Now Pb=75-2*13.75=47.5

Pa=200-25*3.21-2*13.75=92.25