A monopoly is considering selling several units of a homogeneous product as a single package. A typical consumer’s demand for the product is Qd = 50 - 0.25P, and the marginal cost of production is $120.
a. Determine the optimal number of units to put in a package.
b. How much should the firm charge for this package?
Answer a) Given,
Q = 50 - 0.25P
=> 0.25P = 50 - Q
=> P = (50 - Q) / 0.25
=> P = 200 - 4Q
TR (Total Revenue) = P * Q = (200 - 4Q) * Q
=> TR = 200Q - 4Q^2
MR (Marginal Revenue) = TR /
Q
=> MR = 200 - 8Q
MC (Marginal Cost) = $120
For monopolist the profit-maximizing condition is MR = MC. So,
200 - 8Q = 120
=> 200 - 120 = 8Q
=> 8Q = 80
=> Q = 80 / 8
=> Q = 10
Here the optimal number of units for a package is 10 units.
b) From the demand function we get,
P = 200 - 4Q
=> P = 200 - (4 * 10)
=> P = $160
Therefore, here the firm should charge $160 for the package.
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