Jacko's rock band is putting out a new CD with its music label. 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 record label projects the demand for the album 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. b. Compute the profit maximizing price that the label will wish to set. c. What price will Jacko want his band's CD sold for? (Assume he only cares about money earned from the CDs.)
a. The joint profit function is:
Π = (50-0.003Q)Q - 5Q
The optimum is where dΠ/dQ = 0, or:
50 - 0.006Q -5 = 0
Q = 7500, p = 27.5
b. The label's net profits are:
Π = .75(50 - 0.003Q)Q - 5Q
The first order condition is:
37.5 - .0045Q - 5 = 0
Solving Q = 7222, p = 28.33
c. Jeremy's income is:
Π = .25(50 - 0.003Q)Q + 10,000
The 10,000 does not influence the optimal price and quantity for Jacko, which occurs where:
12.5 - 0.0015Q = 0
or Q = 8,333 and p = 25.
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