The profit-maximising output

Overview

Total revenue.

What you need to know

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Questions from this subtopic

Each one with the model answer underneath.

1. 1

   What condition must hold at the margin for profit to be maximised, and why does it hold?

   Model answer

   Profit is maximised where marginal revenue equals marginal cost, provided marginal cost is rising through marginal revenue at that point. If marginal revenue exceeds marginal cost, the extra unit adds more to revenue than to cost, so producing it raises profit and the firm should expand. If marginal cost exceeds marginal revenue, the extra unit reduces profit, so the firm should contract. Profit can therefore only be at its maximum where the two are equal.

2. 2

   Why is the condition that marginal revenue equals marginal cost not on its own sufficient for maximum profit?

   Model answer

   The two curves can also be equal where marginal cost is falling and cuts marginal revenue from above, and at such a point the firm is at a minimum of profit rather than a maximum. Expanding output from there would raise profit, since marginal revenue would then exceed the falling marginal cost. The maximum requires marginal cost to be rising through marginal revenue, so the second intersection is the relevant one.

3. 3

   A firm can sell any quantity at 41 pounds. Its marginal costs are 40, 24, 14, 18, 34 and 56 pounds for units one to six. Determine the profit-maximising output and prove it by calculating profit at each output, given total fixed costs of 60 pounds and total variable costs of 40, 64, 78, 96, 130 and 186 pounds.

   Model answer

   Marginal revenue is a constant 41 pounds, and marginal cost first exceeds it on the sixth unit at 56 pounds, so output should be five units. Checking with profit, total revenue is 41, 82, 123, 164, 205 and 246 pounds and total cost is 100, 124, 138, 156, 190 and 246 pounds, giving profit of -59, -42, -15, 8, 15 and 0 pounds. The maximum profit of 15 pounds does indeed occur at five units.

4. 4

   A firm faces P = 100 - 2Q and has costs TC = 30 + 10Q + Q squared. Calculate the profit-maximising output, price and profit.

   Model answer

   Total revenue is 100Q - 2Q squared, so marginal revenue is 100 - 4Q, and marginal cost is 10 + 2Q. Setting 100 - 4Q = 10 + 2Q gives 90 = 6Q, so Q = 15 units. Price is 100 - 2 multiplied by 15 = 70 pounds, total revenue is 70 multiplied by 15 = 1,050 pounds, total cost is 30 + 150 + 225 = 405 pounds and profit is 1,050 - 405 = 645 pounds.

5. 5

   For a firm producing 15 units where P = 100 - 2Q and TC = 30 + 10Q + Q squared, verify that marginal revenue equals marginal cost and that profit falls either side of that output.

   Model answer

   At Q = 15, marginal revenue is 100 - 4 multiplied by 15 = 40 pounds and marginal cost is 10 + 2 multiplied by 15 = 40 pounds, so the condition holds. At Q = 14 profit is 1,008 - 366 = 642 pounds and at Q = 16 profit is 1,088 - 446 = 642 pounds, both below the 645 pounds earned at 15 units. Profit therefore peaks exactly where the two margins are equal.

6. 6

   What would a diagram of profit maximisation look like for a firm with market power?

   Model answer

   Put output on the horizontal axis and price, revenue and cost per unit on the vertical axis, then draw a downward-sloping average revenue curve with a marginal revenue curve twice as steep beneath it, a U-shaped average total cost curve and a marginal cost curve cutting it at its minimum. The profit-maximising output is directly below the point where marginal cost cuts marginal revenue from below, and the price is read vertically up from that output to the average revenue curve. Supernormal profit is the rectangle bounded above by the price, below by average total cost at that output and horizontally by the output itself.

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