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Difficulty: HardRevenue Concepts and Relationships

A firm operating in an imperfectly competitive market faces a demand function given by P=160−4QP = 160 - 4Q, where PP is the unit price in Naira (₦\text{₦}) and QQ is the output quantity in units. At what output level QQ will the firm maximize its Total Revenue (TRTR)?

Answer: 20 units

Answer

The firm maximizes its total revenue at an output level of 20 units.
Total revenue (TRTR) is maximized at the point where additional output yields zero additional revenue, meaning marginal revenue (MRMR) equals zero (MR=0MR = 0). Given the demand function P=160−4QP = 160 - 4Q, total revenue is TR=P×Q=160Q−4Q2TR = P \times Q = 160Q - 4Q^2. The marginal revenue function is MR=dTRdQ=160−8QMR = \frac{dTR}{dQ} = 160 - 8Q. Setting MR=0MR = 0 gives 160−8Q=0160 - 8Q = 0, which solves to Q=20Q = 20 units.

Step-by-Step Solution

1
Derive the Total Revenue (TR) function from the demand function.
TR=160Q−4Q2TR = 160Q - 4Q^2
Total revenue is calculated by multiplying price (PP) by quantity (QQ).
2
Derive the Marginal Revenue (MR) function.
MR=160−8QMR = 160 - 8Q
Marginal revenue is the rate of change of total revenue with respect to output (dTRdQ\frac{dTR}{dQ}).
3
Set Marginal Revenue to zero and solve for output (QQ).
8Q=160  ⟹  Q=208Q = 160 \implies Q = 20
Total revenue reaches its maximum peak when marginal revenue declines to zero (MR=0MR = 0).

Key Concept

Total Revenue Maximization Condition (MR=0MR = 0)
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Topics

Revenue Concepts and Relationships

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