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

A monopolist faces a market demand curve given by P=100−4QP = 100 - 4Q, where PP is price in Naira and QQ is output quantity. The firm operates with a constant marginal cost MC=N20MC = \text{N}20 and total fixed costs of N100\text{N}100. What are the profit-maximizing total revenue and economic profit for this firm in the short run?

  1. Total revenue is N600\text{N}600 and economic profit is N300\text{N}300Answer
  2. B
    Total revenue is N400\text{N}400 and economic profit is −N100-\text{N}100
  3. C
    Total revenue is N625\text{N}625 and economic profit is N275\text{N}275
  4. D
    Total revenue is N300\text{N}300 and economic profit is N0\text{N}0

Answer

Total revenue is N600\text{N}600 and economic profit is N300\text{N}300
To maximize profit, the monopolist equates MRMR to MCMC. Given demand P=100−4QP = 100 - 4Q, total revenue is TR=100Q−4Q2TR = 100Q - 4Q^2, yielding marginal revenue MR=100−8QMR = 100 - 8Q. Equating MR=20MR = 20 yields Q=10Q = 10 units. Substituting Q=10Q = 10 into the demand function gives price P=N60P = \text{N}60. Thus, total revenue is 60×10=N60060 \times 10 = \text{N}600. Total cost is TFC+TVC=100+20(10)=N300TFC + TVC = 100 + 20(10) = \text{N}300. Subtracting total cost from total revenue yields an economic profit of N300\text{N}300.

Step-by-Step Solution

1
Derive Total Revenue (TR) and Marginal Revenue (MR) functions
Total revenue TR=P×Q=(100−4Q)Q=100Q−4Q2TR = P \times Q = (100 - 4Q)Q = 100Q - 4Q^2. Differentiating with respect to QQ gives MR=100−8QMR = 100 - 8Q.
Monopolists face a downward-sloping demand curve, so marginal revenue lies below the price line.
2
Equate Marginal Revenue to Marginal Cost (MR=MCMR = MC) to find profit-maximizing output level (QQ)
100−8Q=20  ⟹  8Q=80  ⟹  Q=10100 - 8Q = 20 \implies 8Q = 80 \implies Q = 10 units.
The profit-maximizing condition for any firm, including a monopoly, is MR=MCMR = MC.
3
Calculate the market price (PP) and Total Revenue (TRTR)
P=100−4(10)=N60P = 100 - 4(10) = \text{N}60. TR=60×10=N600TR = 60 \times 10 = \text{N}600.
The monopolist sets price according to the market demand curve at the profit-maximizing output quantity.
4
Calculate Total Cost (TCTC) and Economic Profit (π\pi)
TC=TFC+TVC=100+20(10)=N300TC = TFC + TVC = 100 + 20(10) = \text{N}300. Profit π=TR−TC=600−300=N300\pi = TR - TC = 600 - 300 = \text{N}300.
Economic profit is the difference between total revenue earned and total production costs incurred.

Key Concept

Monopoly Short-Run Profit Maximization
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Topics

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

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