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Difficulty: MediumMonopoly: Short-Run and Long-Run Price and Output Determination

A single-price monopolist operates with a total revenue function given by TR=120Q−3Q2TR = 120Q - 3Q^2 and a total cost function given by TC=100+20Q+2Q2TC = 100 + 20Q + 2Q^2, where QQ represents the output quantity in units and figures are in Naira (₦). What is the profit-maximizing price charged by the firm?

  1. ₦90Answer
  2. B
    ₦60
  3. C
    ₦40
  4. D
    ₦20

Answer

The profit-maximizing price is ₦90.
To maximize profits, a monopolist produces where marginal revenue equals marginal cost (MR=MCMR = MC). Differentiating total revenue TR=120Q−3Q2TR = 120Q - 3Q^2 yields MR=120−6QMR = 120 - 6Q, and differentiating total cost TC=100+20Q+2Q2TC = 100 + 20Q + 2Q^2 yields MC=20+4QMC = 20 + 4Q. Setting 120−6Q=20+4Q120 - 6Q = 20 + 4Q solves to Q=10Q = 10 units. The demand equation for price is P=TRQ=120−3QP = \frac{TR}{Q} = 120 - 3Q. Substituting Q=10Q = 10 gives P=120−3(10)=₦90P = 120 - 3(10) = ₦90.

Step-by-Step Solution

1
Derive the Marginal Revenue (MR) and Marginal Cost (MC) functions.
MR=dTRdQ=120−6QMR = \frac{dTR}{dQ} = 120 - 6Q and MC=dTCdQ=20+4QMC = \frac{dTC}{dQ} = 20 + 4Q.
Profit maximization occurs at the output level where Marginal Revenue equals Marginal Cost.
2
Set MR equal to MC to solve for profit-maximizing output quantity (QQ).
120−6Q=20+4Q  ⟹  100=10Q  ⟹  Q=10120 - 6Q = 20 + 4Q \implies 100 = 10Q \implies Q = 10 units.
Equating MR and MC identifies the specific output level that maximizes total profit.
3
Determine the Average Revenue (Demand) equation and solve for Price (PP).
P=TRQ=120−3QP = \frac{TR}{Q} = 120 - 3Q. Substituting Q=10Q = 10 yields P=120−3(10)=₦90P = 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.

Key Concept

Monopoly Profit Maximization (MR=MCMR = MC and Price Determination)
Estimated Time:1m 30s
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Topics

Monopoly: Short-Run and Long-Run Price and Output Determination

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