Question

Jack’s rock band is putting out a new CD with its music label. The demand for the album is p=50-0.003Q where p is the price per CD (in $) and Q is the number of CDs demanded. The cost (not including the band’s salary) of producing the CD is constant at 5$ per disc.

a) Compute the joint profit-maximizing price and quantity, i.e. the price and quantity that maximizes total profit (Revenue – Total Cost) of producing the CD.

b) The contract between the band and the label specifies that the band receive 25% of the gross revenues plus another 10,000$ up front. The label is responsible for the production costs. Compute the price that maximizes the label’s profit.

c) Given the contract described in part (b), what price will Jack want his band’s CD sold for if he only cares about maximizing the band’s profit?

Answer 1

(a) The total revenue will be or or . The average cost is or . The profit would be or or . The profit maximizing quantity would be where (the FOC), and hence, we have or or or units. The profit maximizing price would be hence dollars.

(b) The total cost in this case would be or or or . The profit would be hence or .

The FOC is hence or or or or 7222 units would be the profit maximizing quantity. The profit maximizing price would be dollars.

(c) The band's total revenue would be , as the band receives 25% of total revenues and $10000. The total cost associated is none, as it is not the band that produces CDs, but the label. The profit is hence the revenue itself, ie or or .

The profit would be maximum where or or or or 8333 units. The price would hence be dollars.