Choosing Output From a Marginal-Cost Schedule

From a Marginal-Cost Schedule to an Output Choice

A price-taking firm chooses how many units to produce, but it may not have a smooth cost curve or a formula for total cost. Instead, it may be given a table: the first unit costs one amount to produce, the second unit adds another amount, and so on. To choose output from that schedule, compare the revenue from each additional unit with the cost of making it.

Assume the firm sells each unit at a fixed market price $P$, as a price-taking firm does. The firm receives $P$ dollars for one more unit, while its marginal-cost schedule states how much the firm’s total cost rises when it makes that unit. The comparison determines whether adding the unit raises or lowers profit. This tutorial focuses on that discrete comparison; it does not require the firm’s cost schedule to be written as a continuous curve.

Definition: For an integer output level $q \geq 1$, the marginal cost of the $q$th unit is $MC_q=C(q)-C(q-1)$, where $C(q)$ is total cost at output $q$. With a constant market price $P$, revenue at output $q$ is $Pq$. The change in profit from increasing output from $q-1$ to $q$ is $P-MC_q$.

The subscript matters: $MC_q$ is the extra cost of moving to output $q$, or equivalently the cost of producing the $q$th unit. It is not the cost of the next unit after $q$; that would be $MC_{q+1}$. If a table labels costs by the unit number, compare the price to the cost listed for each unit.

Why the Unit-by-Unit Comparison Works

Let profit at output $q$ be $\pi(q)=Pq-C(q)$. Moving from $q-1$ units to $q$ units changes revenue by $Pq-P(q-1)=P$. It changes total cost by $C(q)-C(q-1)=MC_q$. Subtracting the change in cost from the change in revenue gives the change in profit:

\pi(q)-\pi(q-1) =\bigl[Pq-C(q)\bigr]-\bigl[P(q-1)-C(q-1)\bigr] =P-\bigl[C(q)-C(q-1)\bigr] =P-MC_q.

This argument uses a constant price for the firm’s output and a schedule that correctly records the incremental cost of each unit. It does not depend on whether the firm has a fixed cost: a fixed cost is the same at $q-1$ and $q$, so it cancels from the incremental cost. Fixed costs can affect the level of profit, but not this comparison between adjacent output levels.

If marginal cost rises as output increases, the decision is especially direct. Produce units for which price exceeds marginal cost; a unit with price below marginal cost reduces profit. If price equals the marginal cost of a unit, adding that unit leaves profit unchanged, so the two adjacent output levels tie. When several consecutive units have marginal cost equal to price, all output levels across those zero-change units have the same profit.

Rule for a rising marginal-cost schedule: Increase output while the next unit’s marginal cost is below the market price. Stop before the first unit whose marginal cost exceeds price. If a unit’s marginal cost exactly equals price, producing it is optional on profit grounds: output with and without that unit gives the same profit.

The rising-cost condition is important when using a simple cutoff. If marginal costs do not rise, an early unit may have a cost above price while a later unit has a cost far below price. The firm cannot produce the later unit without also producing earlier units, so it must compare the total profit change across feasible output levels. The cumulative method works whether or not marginal cost rises.

Read the schedule in unit order.
Identify the marginal cost of each additional unit, not an average cost or the cost of the preceding unit.

Calculate each unit’s incremental profit.
For unit $q$, calculate $P-MC_q$. A positive result raises profit, zero leaves it unchanged, and a negative result lowers it.

Choose a feasible output and verify it.
With rising marginal cost, use the cutoff rule. With an irregular schedule, accumulate the changes from zero output and select the output with the largest cumulative profit, including any fixed cost if total profit is required.

Worked Decisions with a Rising Schedule

Worked Example: A Unit Exactly at the Market Price

A hypothetical workshop sells each finished component for $P=\$18$. Its marginal costs for the first five units are $\$6,\$10,\$14,\$18,$ and $\$22$, respectively. The workshop also pays a fixed cost of $\$20$, whether it produces or not. Choose the profit-maximizing output and verify the result using total profit.

Method. Compare the price with the marginal cost of each unit. Since the schedule is increasing, a positive incremental profit is followed by zero or negative increments once the schedule reaches the price.

The changes in profit from units one through five are:

\begin{aligned} P-MC_1 &= 18-6=12,\\ P-MC_2 &= 18-10=8,\\ P-MC_3 &= 18-14=4,\\ P-MC_4 &= 18-18=0,\\ P-MC_5 &= 18-22=-4. \end{aligned}

The first three units each increase profit. The fourth adds no profit, and the fifth lowers profit by $\$4$. Thus output of three or four units is optimal. To check, the variable cost of three units is $6+10+14=\$30$; revenue is $3(18)=\$54$. Total cost including the fixed cost is $30+20=\$50$, so profit is $54-50=\$4$. For four units, variable cost is $30+18=\$48$, revenue is $4(18)=\$72$, and total cost is $48+20=\$68$, again giving profit $72-68=\$4$.

At five units, variable cost is $48+22=\$70$, revenue is $5(18)=\$90$, and total cost is $70+20=\$90$, giving profit $\$0$. This is $\$4$ below the maximum. The fixed cost lowers profit at every output, but does not change which output levels tie for the maximum.

Common mistake avoided: Treating a unit with $MC=P$ as a reason that the firm must stop at the preceding output. The equality means the unit adds zero profit, so both adjacent output choices are optimal here.

Worked Example: Finding the Cutoff and Checking Total Profit

A hypothetical parts maker faces a market price of $P=\$25$ per unit. Its marginal-cost schedule for the first six units is $\$9,\$13,\$17,\$23,\$29,$ and $\$35$. Its fixed cost is $\$30$. Find the maximizing output and calculate the corresponding profit.

Method. Compare each unit’s marginal cost with the price, then use total revenue minus total cost to check the cutoff and its neighboring output levels.

The incremental profit from the first six units is $25-9=\$16$, $25-13=\$12$, $25-17=\$8$, $25-23=\$2$, $25-29=-\$4$, and $25-35=-\$10$. The first four units raise profit; the fifth and sixth lower it. The maximizing output is therefore four units.

For four units, variable cost is $9+13+17+23=\$62$. Total cost is $62+30=\$92$, and revenue is $4(25)=\$100$. Profit is $100-92=\$8$.

Check three units: variable cost is $9+13+17=\$39$, total cost is $39+30=\$69$, revenue is $3(25)=\$75$, and profit is $75-69=\$6$. Check five units: variable cost is $62+29=\$91$, total cost is $91+30=\$121$, revenue is $5(25)=\$125$, and profit is $125-121=\$4$. Four units outperform both neighboring choices.

Common mistake avoided: Stopping after finding that the first four units have positive marginal profit without checking what happens at the next unit. The fifth unit has marginal cost above price, so it reduces profit; the total-profit check confirms the cutoff.

When Marginal Costs Do Not Rise

A schedule may be irregular. In that case, a firm should not automatically stop at the first unit whose marginal cost exceeds price. Because output choices are sequential, the firm must produce earlier units to reach a later one. Calculate the profit change for every unit, add those changes cumulatively from zero output, and compare the cumulative totals.

Worked Example: An Irregular Marginal-Cost Schedule

A hypothetical producer can sell each unit for $P=\$20$. The marginal costs of its first four units are $\$12,\$24,\$8,$ and $\$25$, in that order. Assume there is no fixed cost. Find the output that maximizes profit.

Method. Calculate each unit’s change in profit, then add the changes in sequence. This method respects the requirement that output of three units includes the first and second units.

The incremental profit changes are $20-12=\$8$, $20-24=-\$4$, $20-8=\$12$, and $20-25=-\$5$. Starting from profit of $\$0$ at zero output, cumulative profit is:

\begin{aligned} \pi(0)&=0,\\ \pi(1)&=0+8=8,\\ \pi(2)&=8-4=4,\\ \pi(3)&=4+12=16,\\ \pi(4)&=16-5=11. \end{aligned}

The highest value is $\pi(3)=\$16$, so the maximizing output is three units. A total-cost check gives the same answer: total cost at three units is $12+24+8=\$44$, revenue is $3(20)=\$60$, and profit is $60-44=\$16$. At two units, revenue is $\$40$ and cost is $12+24=\$36$, for profit of $\$4$. At four units, revenue is $\$80$ and cost is $12+24+8+25=\$69$, for profit of $\$11$.

Common mistake avoided: Stopping at one unit because the second unit costs $\$24$, more than the $\$20$ price. That would miss the profitable third unit. Producing three units requires accepting the loss on unit two, but the cumulative profit at three units is still highest.

Reading the Schedule Carefully

A reliable answer states both the output and the reasoning behind it. In particular, be clear about the difference between a unit’s marginal contribution and the firm’s total profit. A unit with $P>MC_q$ raises profit relative to producing one fewer unit, but that fact alone does not show that total profit is positive: prior costs, including fixed cost, still matter. Conversely, a firm can have a negative total profit at its best available output; the marginal schedule identifies which output is best, not whether the firm earns a positive profit.

For a rising schedule, the key comparison is the market price against the marginal cost of the next unit under consideration. If the table lists $MC_1,MC_2,\ldots$, then after choosing $q$ units, the next comparison is $P$ against $MC_{q+1}$. Mixing up $MC_q$ and $MC_{q+1}$ can shift the answer by one unit. For any schedule, a final check of total profit across the best candidate outputs can expose that error.

Key takeaway: At a constant market price, the change in profit from making one more unit is $P-MC_q$. Use the rising-schedule cutoff when marginal costs increase; when the schedule is irregular, accumulate the changes across feasible output levels and choose the largest total profit.

Check Your Understanding

For each problem, identify the output choice and use the stated information to check your reasoning.

A price-taking firm receives $\$16$ per unit. Marginal costs for its first four units are $\$5,\$11,\$16,\$19$, and the schedule is increasing. Which output levels maximize profit? (Answer: 2 or 3 units.)
A firm sells its output for $\$30$ per unit. The marginal costs of its first five units are $\$8,\$17,\$24,\$31,\$36$. How many units maximize profit, and what is the change in profit from adding the fourth unit? (Answer: 3 units; the fourth unit changes profit by $-\$1$.)
A hypothetical firm receives $\$22$ per unit and has marginal costs of $\$10,\$18,\$27,\$34$. It also pays a fixed cost of $\$15$. What output maximizes profit, and what is the profit at that output? (Answer: 2 units; profit is $\$1$.)
A producer’s price is $\$15$, and its first four marginal costs are $\$6,\$19,\$7,\$20$, with no fixed cost. Compute cumulative profit from zero output and identify the maximizing output. (Answer: cumulative profits are $\$0,\$9,\$5,\$13,\$8$; 3 units maximize profit.)
A firm faces a constant price of $\$12$. Its first five marginal costs are $\$12,\$12,\$15,\$9,\$18$, in that order, and it has no fixed cost. Compare total profit at every feasible output from zero to five units. Which output maximizes profit? (Answer: profits are $\$0,\$0,\$0,-\$3,\$0,-\$6$; outputs of 0, 1, 2, or 4 units tie for the maximum.)

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