You are the manager of a monopoly, and your demand and cost functions are given

Question

You are the manager of a monopoly, and your demand and cost functions are given by P= 200 - 2Q and C(Q)= 2000 + 3Q^2, respectively.

a. what price-quantity combination maximizes your frm'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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