A single-price monopolist operates with a total revenue function given by TR = 120Q - 3Q^2… — JAMB UTME

A single-price monopolist operates with a total revenue function given by $TR = 120Q - 3Q^2$ and a total cost function given by $TC = 100 + 20Q + 2Q^2$ , where $Q$ represents the output quantity in units and figures are in Naira (₦). What is the profit-maximizing price charged by the firm?

₦90Cevap
B
₦60
C
₦40
D
₦20

Cevap

The profit-maximizing price is ₦90.

To maximize profits, a monopolist produces where marginal revenue equals marginal cost (

MR = MC

). Differentiating total revenue

TR = 120Q - 3Q^2

yields

MR = 120 - 6Q

, and differentiating total cost

TC = 100 + 20Q + 2Q^2

yields

MC = 20 + 4Q

. Setting

120 - 6Q = 20 + 4Q

solves to

Q = 10

units. The demand equation for price is

P = \frac{TR}{Q} = 120 - 3Q

. Substituting

Q = 10

gives

P = 120 - 3(10) = ₦90

Adım Adım Çözüm

Derive the Marginal Revenue (MR) and Marginal Cost (MC) functions.

MR = \frac{dTR}{dQ} = 120 - 6Q

and

MC = \frac{dTC}{dQ} = 20 + 4Q

Profit maximization occurs at the output level where Marginal Revenue equals Marginal Cost.

Set MR equal to MC to solve for profit-maximizing output quantity (

Q

120 - 6Q = 20 + 4Q \implies 100 = 10Q \implies Q = 10

units.

Equating MR and MC identifies the specific output level that maximizes total profit.

Determine the Average Revenue (Demand) equation and solve for Price (

P

P = \frac{TR}{Q} = 120 - 3Q

. Substituting

Q = 10

yields

P = 120 - 3(10) = ₦90

A monopolist sets its price based on what consumers are willing to pay for the profit-maximizing output according to the demand curve.

Anahtar Kavram

Monopoly Profit Maximization (

MR = MC

and Price Determination)

Tahmini Süre:1m 30s

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