Choosing Output One Unit at a Time
A monopolist chooses the quantity it will sell, but its price is constrained by demand: to sell a different quantity, it generally must charge a different price. The practical decision is therefore not simply “produce where the price is highest.” The firm must compare the additional revenue from expanding output with the additional cost, and then identify the price consumers will pay for the chosen quantity.
This tutorial considers a single-price monopolist: it sells each unit at the same price, cannot set a separate price for each buyer, and faces a demand schedule showing the price needed to sell each possible quantity. Costs are summarized by total cost at each quantity. The examples use discrete units, so marginal amounts refer to the change caused by adding one unit. This approach to monopoly choice is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.
For a discrete schedule, write the marginal revenue from moving from \(Q-1\) units to \(Q\) units as \(MR_Q=TR(Q)-TR(Q-1)\). Similarly, \(MC_Q=TC(Q)-TC(Q-1)\). These are increments for the next unit, not averages across all units. If a problem instead gives a smooth demand or cost curve, marginal revenue and marginal cost describe the change in revenue or cost from a small change in output.
Why Compare Marginal Revenue With Marginal Cost?
The reason for the rule is the change in profit from one more unit. Total profit after adding that unit is higher or lower according to whether the added revenue exceeds the added cost. Subtracting the profit before expansion from the profit after expansion gives:
Thus, if \(MR_Q>MC_Q\), adding the unit increases profit. If \(MR_Q<MC_Q\), adding it decreases profit. If they are equal, adding that unit leaves profit unchanged. In a smooth, unconstrained choice, the familiar candidate for an interior maximum is where marginal revenue equals marginal cost. In a schedule, however, compare the increments for feasible units, and check the resulting profits. A crossing is not a substitute for checking the available choices when schedules are irregular or output is restricted.
For each feasible quantity, multiply that quantity by the demand price. Do not use the price alone as total revenue.
Subtract the previous quantity’s total revenue and total cost from the current quantity’s totals. Each difference applies to the added unit.
Continue adding units while the next unit’s marginal revenue exceeds its marginal cost. A unit with lower marginal revenue than marginal cost reduces profit relative to the preceding quantity.
Use the demand schedule’s price at the selected quantity, then compare total profit across feasible alternatives, especially when the marginal schedules are irregular.
Worked Examples
Worked Example: Finding the Best Quantity From a Schedule
A hypothetical maker of specialized storage cases is the only seller in its market. The table gives the price needed to sell each quantity and the firm’s total cost. Find the profit-maximizing quantity and its price.
| Quantity | Price | Total revenue | Marginal revenue | Total cost | Marginal cost | Profit |
|---|---|---|---|---|---|---|
| 0 | — | $0 | — | $8 | — | −$8 |
| 1 | $18 | $18 | $18 | $12 | $4 | $6 |
| 2 | $16 | $32 | $14 | $20 | $8 | $12 |
| 3 | $14 | $42 | $10 | $31 | $11 | $11 |
| 4 | $12 | $48 | $6 | $45 | $14 | $3 |
| 5 | $10 | $50 | $2 | $63 | $18 | −$13 |
Method: calculate the increments, then compare profits. At quantity 2, total revenue is \(2\times \$16=\$32\), and marginal revenue for the second unit is \(\$32-\$18=\$14\). Total cost is \(\$20\), so marginal cost for that unit is \(\$20-\$12=\$8\). Its addition raises profit by \(\$14-\$8=\$6\).
For the third unit, marginal revenue is \(\$42-\$32=\$10\), while marginal cost is \(\$31-\$20=\$11\). The third unit reduces profit by \(\$1\). The fourth and fifth units also have marginal revenue below marginal cost: \(\$6<\$14\) and \(\$2<\$18\). Profit is highest at quantity 2: \(\$32-\$20=\$12\). The corresponding demand price is \(\$16\).
Check: Profit at quantities 1 through 5 is \(\$6,\$12,\$11,\$3,-\$13\), respectively, and profit at zero output is \(-\$8\). The maximum is indeed \(\$12\) at quantity 2. The comparison avoids the mistake of stopping at the highest demand price, which occurs at quantity 1, rather than choosing the quantity with the highest profit.
Worked Example: Why Price Is Not Marginal Revenue
A hypothetical supplier of laboratory filters faces the demand schedule below. Its total costs are also shown. Determine the best output and price, and explain why comparing the demand price directly with marginal cost gives the wrong result.
| Quantity | Price | Total revenue | Marginal revenue | Total cost | Marginal cost | Profit |
|---|---|---|---|---|---|---|
| 0 | — | $0 | — | $5 | — | −$5 |
| 1 | $30 | $30 | $30 | $17 | $12 | $13 |
| 2 | $26 | $52 | $22 | $34 | $17 | $18 |
| 3 | $22 | $66 | $14 | $55 | $21 | $11 |
| 4 | $18 | $72 | $6 | $81 | $26 | −$9 |
Step 1: Find total revenue. At quantity 2, revenue is \(2\times \$26=\$52\). At quantity 3, it is \(3\times \$22=\$66\).
Step 2: Calculate the incremental revenue and cost. The third unit raises revenue by \(\$66-\$52=\$14\), not by its \(\$22\) selling price. It raises cost by \(\$55-\$34=\$21\). Because \(\$14<\$21\), moving from 2 to 3 units lowers profit by \(\$7\).
Step 3: Select the output and price. The second unit has marginal revenue \(\$22\) and marginal cost \(\$17\), so it increases profit. The third and fourth units each reduce profit. Profit at quantities 0 through 4 is \(-\$5,\$13,\$18,\$11,-\$9\), so the maximum is \(\$18\) at quantity 2. The demand price at that output is \(\$26\).
Check and common mistake: Comparing the price of the third unit, \(\$22\), with its marginal cost, \(\$21\), might suggest expanding to quantity 3. That calculation ignores the revenue lost when the price falls from \(\$26\) to \(\$22\) on the first two units. The correct marginal revenue is \(\$14\), as confirmed by the change in total revenue.
Worked Example: Check the Whole Schedule When Marginals Are Irregular
A hypothetical firm sells replacement components. Demand and cost are listed for each feasible quantity. Find the most profitable output. In particular, explain why a later unit with marginal revenue above marginal cost does not automatically make a larger output best.
| Quantity | Price | Total revenue | Marginal revenue | Total cost | Marginal cost | Profit |
|---|---|---|---|---|---|---|
| 0 | — | $0 | — | $0 | — | $0 |
| 1 | $25 | $25 | $25 | $7 | $7 | $18 |
| 2 | $21 | $42 | $17 | $17 | $10 | $25 |
| 3 | $17 | $51 | $9 | $29 | $12 | $22 |
| 4 | $14 | $56 | $5 | $33 | $4 | $23 |
| 5 | $11 | $55 | −$1 | $36 | $3 | $19 |
The first two units each raise profit: their marginal comparisons are \(\$25>\$7\) and \(\$17>\$10\). The third unit reduces profit by \(\$12-\$9=\$3\), taking profit from \(\$25\) to \(\$22\). The fourth unit then raises profit by \(\$5-\$4=\$1\), but this recovers only part of the earlier loss. At quantity 4, profit is \(\$56-\$33=\$23\), still below the \(\$25\) at quantity 2.
The fifth unit reduces profit by a further \(\$4\), because its marginal revenue is \(-\$1\) and its marginal cost is \(\$3\). Comparing all feasible quantities shows the maximum is at quantity 2, where the demand price is \(\$21\). This is why, with an irregular marginal-cost schedule, a profitable later increment does not establish that the quantity reached is the global maximum.
Check: The listed profits are \(0,\$18,\$25,\$22,\$23,\$19\). Their largest value is \(\$25\), at quantity 2. The method avoids relying on a single local comparison when earlier and later increments do not follow a simple monotonic pattern.
Reading the Decision Correctly
Marginal analysis identifies a quantity; it does not directly give the price. Once the quantity is selected, return to the demand schedule and read the price consumers will pay for that quantity. For example, if the marginal comparison points to quantity 2, the relevant price is the demand price at quantity 2—not the marginal revenue of the second unit.
The same distinction applies to profit. A marginal comparison tells whether one step from a particular quantity raises or lowers profit. Total profit at a quantity is \(TR-TC\), which includes the consequences of every unit produced. If marginal revenue falls and marginal cost rises as quantity increases, the comparison often gives a clear stopping point: units before the crossing add to profit and units after it reduce profit. If either schedule behaves irregularly, or only a limited set of quantities is feasible, check total profit across the available choices.
- Do not substitute price for marginal revenue. For a single-price monopolist, selling an extra unit may require lowering the price on units it was already selling.
- Keep increments aligned. The marginal revenue and marginal cost used in a comparison must both refer to the same added unit.
- Do not confuse marginal profit change with total profit. A unit can raise profit while the resulting output still earns less than another feasible output.
- Read price after selecting quantity. The demand schedule supplies the price associated with the chosen output.
Check Your Understanding
For each problem, calculate or use the marginal amounts and identify the output, its demand price, and the profit where requested.
- A hypothetical seller faces prices of $15, $12, and $9 at quantities 1, 2, and 3. Total costs at quantities 0 through 3 are $4, $10, $18, and $29. Calculate the marginal revenue and marginal cost of the second unit, then find the best quantity and its price. (Answer: Second-unit marginal revenue is $9 and marginal cost is $8; choose quantity 2 at $12, with profit $6.)
- A specialty-product monopolist has prices of $24, $20, $16, and $12 at quantities 1 through 4. Total costs at quantities 0 through 4 are $6, $14, $25, $39, and $57. Find the profit-maximizing quantity and verify it using profit. (Answer: Quantity 2 at $20; maximum profit is $15.)
- A seller’s prices at quantities 1 through 5 are $32, $27, $22, $18, and $14. Total costs at quantities 0 through 5 are $9, $20, $34, $51, $69, and $90. Calculate the relevant marginal amounts and identify the best quantity, price, and profit. (Answer: Quantity 2 at $27; maximum profit is $20.)
- A hypothetical equipment supplier faces prices of $50, $42, $34, and $27 at quantities 1 through 4. Its total costs at quantities 0 through 4 are $12, $30, $55, $82, and $115. Determine the best output and price, checking profit at all feasible quantities. (Answer: Quantity 2 at $42; maximum profit is $29.)
- For quantities 1 through 4, a niche seller’s demand prices are $16, $13, $10, and $7. Total costs at quantities 0 through 4 are $0, $5, $12, $20, and $25. Find the best quantity and explain briefly why the fourth unit’s marginal cost alone does not determine the decision. (Answer: Quantity 2 at $13, with profit $14; the fourth unit’s marginal revenue is −$2 and its marginal cost is $5, so it lowers profit from quantity 3.)