You are the manager of a monopoly, and your demand and cost functions are given
ID: 1249054 • Letter: Y
Question
You are the manager of a monopoly, and your demand and cost functions are given by P = 200 - 2Q and C(Q) = 2000 + 3Qsquared, respectively.a. What price-quantity combination maximizes your firm's profits?
b. Calculate the maximum profits.
c. Is demand elastic, inelastic, or unit elastic at the profit-maximizing price-quantity combination?
d. What price-quantity combination maximizes revenue?
e. Calculate the maximum revenues.
f. Is demand elastic, inelastic, or unit elastic at the revenue-maximizing price-quantity combination?
Explanation / Answer
(a) R(Q) = P * Q = (200 - 2Q) * Q = 200Q - 2Q^2
Profit = R(Q) - C(Q) = 200Q - 2Q^2 - (2000 + 3Q^2) = 200Q - 5Q^2 - 2000
For maximum profit, the derivative should be 0
200 - 10Q = 0
Q = 20 units
P = 200 - 2(20) = $160
(b) Maximum profit = 200(20) - 5(20^2) - 2000 = 0
(c) Price elasticity of demand = |(dQ/dP)(P/Q)| = |-2 * (200 - 2Q)/Q| = |-2(200 - 2 * 20)/20| = 16
Since 16 > 1, the demand is price elastic at the profit maximizing quantity and price.
(a) R(Q) = P * Q = (200 - 2Q) * Q = 200Q - 2Q^2
Profit = R(Q) - C(Q) = 200Q - 2Q^2 - (2000 + 3Q^2) = 200Q - 5Q^2 - 2000
For maximum profit, the derivative should be 0
200 - 10Q = 0
Q = 20 units
P = 200 - 2(20) = $160
(b) Maximum profit = 200(20) - 5(20^2) - 2000 = 0
(c) Price elasticity of demand = |(dQ/dP)(P/Q)| = |-2 * (200 - 2Q)/Q| = |-2(200 - 2 * 20)/20| = 16
Since 16 > 1, the demand is price elastic at the profit maximizing quantity and price.
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