A monopolist faces the following inverse demand curve: P= (36-2Q)z; where P is p
ID: 1123640 • Letter: A
Question
A monopolist faces the following inverse demand curve: P= (36-2Q)z; where P is price; Q is her total output; and z is the quality of product sold. z can take on only one of two values. The monopolist can choose to maket a low-quality product for which z=1. Alternatively, the monopolist can choose to market a high-quality product for which z=2. Marginal cost is independent of quality and is constant at zero. Fixed cost, however, depends on the product design and increases with the quality chosen. Specifically, fixed cost is equal to 65z2.
a) find the monopolist's profits if profits are maximized and a low-quality design is chosen.
b) find the monopolist's profits if profits are maximized and a high-quality design is chosen.
Explanation / Answer
In each case, profit is maximized when Marginal revenue (MR) equals Marginal cost (MC), where MC = 0
Profit = Total revenue (TR) - Total cost (TC) = TR - Fixed cost - Variable cost = TR - 65z2 - 0 = TR - 65z2
(a) For low-quality design, z = 1
P = 36 - 2Q
TR = P x Q = 36Q - 2Q2
MR = dTR/dQ = 36 - 4Q
Equating with MC,
36 - 4Q = 0
4Q = 36
Q = 9
P = 36 - (2 x 9) = 36 - 18 = 18
TR = 18 x 9 = 162
Profit = 162 - (65 x 1 x 1) = 162 - 65 = 97
(b) For high-quality design, z = 2
P = (36 - 2Q) x 2 = 72 - 4Q
TR = P x Q = 72Q - 4Q2
MR = dTR/dQ = 72 - 8Q
Equating with MC,
72 - 8Q = 0
8Q = 72
Q = 9
P = 72 - (4 x 9) = 72 - 36 = 36
TR = 36 x 9 = 324
Profit = 324 - (65 x 2 x 2) = 324 - 260 = 64