Choosing Output From A Profit Table

A firm may have only a limited set of production quantities available in the period being considered. In that case, its decision is a comparison among those specific choices: calculate the profit associated with each feasible quantity, then select the quantity or quantities with the greatest profit. The central discipline is to compare profit, not revenue or output alone.

The method is simple, but tables create opportunities for errors: a cost may be subtracted from the wrong row, a fixed cost may be omitted, or a candidate may be chosen because it has the highest revenue even though its costs are also high. Careful row-by-row calculation makes the decision transparent.

Definition: For an output quantity $q$, total profit is total revenue minus total cost: $$ \pi(q)=TR(q)-TC(q). $$ If the feasible output choices are $q_1,q_2,\ldots,q_n$, a profit-maximizing output is any feasible choice $q_j$ whose profit is at least as large as the profit at every other feasible choice. If several choices share the greatest profit, each is profit-maximizing.

The Discrete-Choice Model

Use this comparison when the firm is choosing among a stated, finite set of output quantities for one decision period. For each quantity, the table must pair the total revenue and total cost that correspond to that same quantity and period. The firm is assumed to seek the greatest profit among the feasible choices, and the table is treated as giving the relevant values for the decision. Any production limits or other restrictions should already be reflected in which quantities are listed as feasible.

This is a discrete choice: only the listed quantities are candidates. If the table lists output of 0, 10, and 20 units, it does not establish that output of 13 units is feasible or tell us its profit. Do not interpolate between rows unless the problem supplies a rule that supports doing so. The table also does not imply that an unlisted quantity is feasible.

The procedure follows directly from the definition of profit. At each feasible quantity, subtract the cost incurred at that quantity from the revenue earned at that quantity. The greatest resulting net amount is the best choice under the stated objective. If two rows have the same greatest amount, neither is better by the profit criterion; both attain the maximum.

q^* \in \operatorname*{arg\,max}_{q\in Q}\left[TR(q)-TC(q)\right],

Here $Q$ is the set of feasible output choices, and $q^*$ denotes a profit-maximizing choice. The notation allows more than one answer when the maximum is tied. In a table, the calculation is to complete the profit column and identify its largest entry or entries. This direct method is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Confirm the choices and units.
Read which quantities are feasible and whether they are measured in individual units, batches, or another unit.

Match each row's revenue and cost.
Use total revenue and total cost from the same output level and the same time period.

Calculate profit for every row.
Subtract total cost from total revenue, preserving negative values if costs exceed revenue.

Compare the completed profit column.
Choose every feasible output level with the greatest profit, and report all tied choices.

Worked Comparisons

Worked Example: Selecting the Highest-Profit Output

A hypothetical workshop can produce between zero and five custom display stands in a shift. The table gives total revenue and total cost in dollars for each feasible quantity. Which output maximizes profit?

Stands, $q$	Total revenue, $TR(q)$	Total cost, $TC(q)$
0	$0	$24
1	$36	$40
2	$68	$54
3	$96	$72
4	$120	$94
5	$140	$120

Method. Calculate $TR(q)-TC(q)$ for each listed quantity. At $q=0$, profit is $0-24=-24$ dollars. At $q=1$, it is $36-40=-4$ dollars. At $q=2$, it is $68-54=14$ dollars. At $q=3$, it is $96-72=24$ dollars. At $q=4$, it is $120-94=26$ dollars. At $q=5$, it is $140-120=20$ dollars.

The profit sequence is $-24,-4,14,24,26,20$, in dollars. Its greatest value is $26, at $q=4$. Therefore, the profit-maximizing output is four stands per shift. As a check, the selected row gives revenue of $120 and cost of $94, and $120-94=26$.

Common mistake avoided: Choosing five stands because that row has the highest revenue. Its $140 revenue is accompanied by $120 in cost, leaving $20 profit, which is less than the $26 profit at four stands.

Worked Example: A Tie at Adjacent Feasible Quantities

A hypothetical maker of insulated containers can produce 0, 1, 2, 3, or 4 containers in a short production run. Total revenue and total cost are shown below, in dollars. Identify every profit-maximizing output.

Containers, $q$	Total revenue, $TR(q)$	Total cost, $TC(q)$
0	$0	$16
1	$20	$29
2	$38	$35
3	$54	$51
4	$68	$70

At $q=0$, profit is $0-16=-16$ dollars. At $q=1$, it is $20-29=-9$ dollars. At $q=2$, it is $38-35=3$ dollars. At $q=3$, it is $54-51=3$ dollars. At $q=4$, it is $68-70=-2$ dollars.

The greatest profit is $3, and it occurs at both $q=2$ and $q=3$. The two adjacent feasible quantities are therefore both profit-maximizing. The equality is exact: each row has revenue exceeding cost by $3. There is no basis in the table for claiming that some intermediate, fractional quantity is also feasible.

Method. Complete every profit calculation before deciding whether there is a tie. Common mistake avoided: Reporting only $q=3$ because it is the larger tied quantity, or treating the two equal entries as evidence that the table has a single maximum. The stated objective gives equal profit at both feasible choices, so both belong in the answer.

Worked Example: Output Units and a Full Table Comparison

A hypothetical print shop can run 0, 1, 2, 3, or 4 batches of promotional labels in one afternoon. Each batch contains 100 labels. The table reports total revenue and total cost in dollars for the afternoon. Find the profit-maximizing number of batches, and state the corresponding number of labels.

Batches, $q$	Total revenue	Total cost
0	$0	$30
1	$48	$55
2	$92	$74
3	$126	$99
4	$148	$130

For zero batches, profit is $0-30=-30$ dollars. For one batch, profit is $48-55=-7$ dollars. For two batches, profit is $92-74=18$ dollars. For three batches, profit is $126-99=27$ dollars. For four batches, profit is $148-130=18$ dollars.

The profit values are $-30,-7,18,27,18$ dollars. The maximum is $27 at three batches. Since each batch contains 100 labels, the selected output is $3\times100=300$ labels. Checking the chosen row, $126-99=27$, confirming the reported profit.

Common mistake avoided: Giving the answer as three labels rather than three batches, or comparing a revenue amount in dollars with an output quantity. The table's quantity unit and the money unit describe different things; state each separately.

Reading the Maximum Carefully

A profit table can include losses as well as gains. A negative entry means total cost exceeds total revenue at that quantity; it should remain negative rather than being set to zero. If every listed profit is negative, the greatest entry is still the best of the listed choices under the stated model. Whether a firm can instead stop production or take another action depends on whether that alternative is included among the feasible choices. Do not silently add an option the problem has not specified.

Likewise, the output with the greatest total revenue need not maximize profit. Revenue alone ignores the cost required to produce that output. The output with the greatest profit may also differ from the output with the lowest cost: low cost does not by itself establish that revenue minus cost is greatest. Calculate the same net measure across all rows.

Costs that apply to every output choice still belong in the profit entries if the table includes them. For example, a fixed charge can make profits lower at all quantities without changing which row has the highest profit, but omitting it would give incorrect profit amounts and could affect whether profits are positive or negative. Use the total cost stated for each row rather than replacing it with a selected cost component.

Keep monetary units and output units distinct, and use the precision of the supplied data. If the table gives whole dollars, compare those values as given; do not introduce extra decimal places or round intermediate calculations. When the maximum is tied, report each maximizing feasible quantity and the shared profit. A complete response identifies both the selected quantity or quantities and the maximum profit.

Key takeaway: In a discrete profit table, calculate $TR-TC$ for every feasible output, compare the resulting profits, and report every quantity attaining the greatest value. Equal maximum entries are a genuine tie, not a reason to choose one row arbitrarily.

Check Your Understanding

For each problem, calculate profit at every listed feasible output before identifying the maximum.

A hypothetical bakery can produce 0, 1, 2, or 3 specialty cakes. Total revenue is $0, $25, $46, and $60, respectively; total cost is $14, $30, $40, and $58. Which output maximizes profit? (Answer: 2 cakes, with maximum profit of $6.)
A small workshop can choose 0, 2, 4, 6, or 8 units. Total revenue is $0, $44, $80, $108, and $128; total cost is $20, $40, $62, $90, and $124. Which feasible outputs maximize profit? (Answer: 4 and 6 units, each with profit of $18.)
A hypothetical firm has feasible outputs of 0, 1, 2, and 3 batches. Total revenue is $0, $35, $66, and $90; total cost is $25, $42, $57, and $86. Find the profit-maximizing output and its profit. (Answer: 2 batches, with profit of $9.)
A label printer can produce 0, 1, 2, 3, or 4 rolls in a shift. Total revenue is $0, $32, $60, $81, and $96; total cost is $18, $35, $48, $67, and $90. Identify the maximizing output and check the selected row's profit. (Answer: 3 rolls; profit is $81-$67=$14.)
A hypothetical producer can choose 0 through 5 crates. Total revenue is $0, $50, $94, $132, $160, and $180, while total cost is $22, $49, $76, $103, $132, and $168. Find the maximum profit and the corresponding output. (Answer: 3 crates, with maximum profit of $29.)

Choosing Output From A Profit Table

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