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

A monopolist faces a market demand function given by P=140−2QP = 140 - 2Q, where PP is the price in Naira and QQ is the output level. The total cost function of the firm is TC=20Q+Q2+200TC = 20Q + Q^2 + 200. What is the maximum economic profit, in Naira, earned by the monopolist at equilibrium?

Answer: 1000 Naira

Answer

The maximum economic profit earned by the monopolist at profit-maximizing equilibrium is 1000 Naira.
To maximize economic profit, a monopolist sets marginal revenue equal to marginal cost (MR=MCMR = MC). From P=140−2QP = 140 - 2Q, TR=140Q−2Q2TR = 140Q - 2Q^2, giving MR=140−4QMR = 140 - 4Q. Differentiating TC=20Q+Q2+200TC = 20Q + Q^2 + 200 gives MC=20+2QMC = 20 + 2Q. Equating MR=MCMR = MC yields 140−4Q=20+2Q  ⟹  Q=20140 - 4Q = 20 + 2Q \implies Q = 20 units. Substituting Q=20Q = 20 into the demand equation gives price P=100P = 100 Naira. Total revenue is 20002000 Naira (100×20100 \times 20) and total cost is 10001000 Naira (20(20)+202+20020(20) + 20^2 + 200). The resulting maximum economic profit is 2000−1000=10002000 - 1000 = 1000 Naira.

Step-by-Step Solution

1
Derive Total Revenue (TR) and Marginal Revenue (MR) functions
TR=140Q−2Q2TR = 140Q - 2Q^2 and MR=140−4QMR = 140 - 4Q
Marginal revenue is the first derivative of total revenue with respect to quantity.
2
Derive Marginal Cost (MC) function
MC=20+2QMC = 20 + 2Q
Marginal cost is the first derivative of total cost with respect to quantity.
3
Equate MR to MC to solve for the profit-maximizing output level (Q)
140−4Q=20+2Q  ⟹  6Q=120  ⟹  Q=20140 - 4Q = 20 + 2Q \implies 6Q = 120 \implies Q = 20 units
The necessary condition for profit maximization in all market structures is MR=MCMR = MC.
4
Determine the equilibrium price (P) from the demand curve
P=140−2(20)=100P = 140 - 2(20) = 100 Naira
Monopolists set price based on consumer willingness to pay at the profit-maximizing output level.
5
Calculate Total Revenue (TR), Total Cost (TC), and Economic Profit (\pi)
TR=100×20=2000TR = 100 \times 20 = 2000, TC=20(20)+(20)2+200=1000TC = 20(20) + (20)^2 + 200 = 1000, Profit =2000−1000=1000= 2000 - 1000 = 1000 Naira
Economic profit is the difference between total revenue and total cost at equilibrium output.

Key Concept

Monopoly Profit Maximization Condition (MR = MC)
Estimated Time:2m 30s
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Topics

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

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