Question

Suppose that demand for rollerblades is given by D(p) = A − p. The cost function for all firms is C(y) = wy2 + f, where f is a fixed set-up cost. The marginal cost of production is MC(y) = 2wy

Suppose that the market for rollerblades is now monopolized with A = 100, w = $4, and f = $100. What is the profit-maximizing quantity? What are monopoly profits?

Answer 1

y = A - p = 100 - p
C(y) = wy² + f = 4y² + 100
MC = 2wy = 2(4y) = 8y

Monopolist maximizes profit according to the rule: MR = MC.
y = 100 - p
So, p = 100 - y
Total Revenue, TR = p*y = (100 - y)*y = 100y - y²
Marginal Revenue, MR = d(TR)/dy = 100 - 2y

Now, MR = MC gives,
100 - 2y = 8y
So, 8y + 2y = 10y = 100
So, y = 100/10
So, y = 10

p = 100 - y = 100 - 10 = 90

C = 4(10)² + 100 = 400 + 100 = 500

Profit = TR - C = p*y - 500 = (90*10) - 500 = 900 - 500 = 400

Thus, profit = 400