The Incremental Profit Test

Why Compare One More Unit?

A firm choosing output does not need to reconsider every cost and receipt from the beginning each time it considers an additional unit. It can ask a narrower question: how much will revenue change, and how much will cost change, if output increases by one unit? The difference between those changes is the effect on profit. This comparison is useful when output choices are discrete, such as whole items, service contracts, or production batches.

The incremental profit test focuses on the move from one feasible output to the next. It does not, by itself, establish that a candidate output is best among every possible choice. To find the best output from a sequence, track the changes and compare the total profits at the feasible choices.

Definition: Let $q$ be the firm's output, $TR(q)$ its total revenue, and $TC(q)$ its total cost. Profit at output $q$ is $\pi(q)=TR(q)-TC(q)$. For a proposed increase from $q$ to $q+1$, incremental profit is $\Delta\pi=\pi(q+1)-\pi(q)=\Delta TR-\Delta TC$, where $\Delta TR=TR(q+1)-TR(q)$ and $\Delta TC=TC(q+1)-TC(q)$. If $\Delta\pi>0$, the additional unit raises profit; if $\Delta\pi<0$, it lowers profit; if $\Delta\pi=0$, the two output choices give equal profit.

The comparison is about changes, not whether the firm's total revenue exceeds its total cost at either output. A firm can have a loss at both output levels and still improve its profit by producing one more unit. Conversely, it can have positive profit at both levels while the additional unit reduces that profit.

Assume that the firm compares feasible output levels and that the stated revenue and cost changes include all effects that differ between those levels. Costs that remain unchanged across the comparison do not affect the result. If a fixed cost changes when the firm expands output, however, that change belongs in $\Delta TC$. This is a short-run incremental comparison; it is not a decision about whether the firm should enter or leave a market.

Why the Test Works

The result follows directly from the definition of profit. Subtract profit at the current output from profit at the higher output:

\begin{aligned} \Delta\pi &=\pi(q+1)-\pi(q)\\ &=[TR(q+1)-TC(q+1)]-[TR(q)-TC(q)]\\ &=[TR(q+1)-TR(q)]-[TC(q+1)-TC(q)]\\ &=\Delta TR-\Delta TC. \end{aligned}

Thus the extra unit's effect on profit is positive precisely when its added revenue exceeds its added cost. The derivation does not require revenue or cost to have any particular shape; it requires only that the two output choices have defined revenue and cost values. The rule concerns a finite change between output levels. It should not be confused with a condition based on derivatives for a continuous choice of output.

Incremental profit test: For a move to a higher feasible output, compare the increase in total revenue with the increase in total cost. Producing the additional unit improves profit exactly when $\Delta TR>\Delta TC$. Equality means equal profit at the two adjacent choices; it does not imply that either choice is best among all feasible outputs.

Applying the Test to One Additional Unit

A reliable calculation keeps the starting and ending output levels explicit. First find each change by subtracting the starting value from the ending value. Then subtract the cost change from the revenue change. A negative $\Delta TC$ is possible in some settings, so retain the signs rather than treating every change as a positive amount.

Worked Example: Evaluating a Fifth Unit

A hypothetical workshop can produce integer quantities of a component. At output 4, total revenue is $288 and total cost is $220. At output 5, total revenue is $330 and total cost is $264. Should the workshop produce the fifth unit if its objective is to raise profit?

Method. Compare the changes from output 4 to output 5. The relevant assumptions are that these are the firm's complete revenues and costs at each output and that the figures use the same period and dollar units.

The revenue change is $330-288=\$42$. The cost change is $264-220=\$44$. Therefore,

\Delta\pi=\Delta TR-\Delta TC=42-44=-\$2.

The fifth unit reduces profit by $2, so the workshop should remain at output 4 when choosing between these two levels. Check using total profit: at output 4, $\pi(4)=288-220=\$68$; at output 5, $\pi(5)=330-264=\$66$. The difference is $66-68=-\$2$, as calculated.

Common mistake avoided. Seeing that total revenue at output 5 exceeds total cost, $330>264$, does not show that the fifth unit is worthwhile. That comparison says output 5 has positive profit; it does not compare its profit with output 4.

Following Incremental Profit Across a Sequence

When several output increases are possible, calculate the change for each move. If every successive increment adds profit up to one point and every increment after that reduces profit, the transition identifies the highest-profit choice in the sequence. But increments need not follow that pattern. A positive increment after a negative one can make a later output preferable to an earlier output. For a sequence with irregular changes, calculate or reconstruct total profit at every feasible choice and compare the totals.

Set the comparison.
List the feasible output levels in order, and identify the current and proposed output for each increment.

Find the revenue and cost changes.
For each move, subtract the starting total from the ending total for revenue and for cost.

Calculate the change in profit.
Subtract the cost change from the revenue change, keeping any negative values and units consistent.

Compare feasible choices.
Use the signs to interpret each move; if the signs are irregular or the question asks for the best output, compare total profit across the relevant choices.

Worked Example: Finding the Best Output in an Irregular Sequence

A hypothetical print shop can select one of five production levels, $q=0,1,2,3,4$. Its total revenue and total cost, in dollars for the same production period, are shown below. Assume all listed costs are relevant to the stated output and the shop seeks the highest profit among these five choices.

Output $q$	Total revenue	Total cost	Profit	Change in profit from prior output
0	$0	$12	−$12	—
1	$50	$54	−$4	+$8
2	$98	$105	−$7	−$3
3	$145	$146	−$1	+$6
4	$190	$200	−$10	−$9

Method. Compute profit as total revenue minus total cost at each output, then check each adjacent change using $\Delta TR-\Delta TC$.

At $q=0$, profit is $0-12=-\$12$. From 0 to 1, revenue rises by $50-0=\$50$ and cost rises by $54-12=\$42$, so $\Delta\pi=50-42=+\$8$. Profit at 1 is $-\$4$.

From 1 to 2, revenue rises by $98-50=\$48$, while cost rises by $105-54=\$51$. Thus $\Delta\pi=48-51=-\$3$, and profit falls from $-\$4$ to $-\$7$.

From 2 to 3, revenue rises by $145-98=\$47$, and cost rises by $146-105=\$41$. Thus $\Delta\pi=47-41=+\$6$, taking profit from $-\$7$ to $-\$1$.

From 3 to 4, revenue rises by $190-145=\$45$, and cost rises by $200-146=\$54$. Thus $\Delta\pi=45-54=-\$9$, taking profit from $-\$1$ to $-\$10$.

The highest profit is $-\$1$, at $q=3$. It is still a loss, but it is less negative than the profit at any other listed choice. The change signs are irregular, so stopping at the first negative increment would miss the better choice at $q=3$.

Check. The four changes sum to $8-3+6-9=+\$2$. Starting from $-\$12$, the resulting profit at $q=4$ is $-12+2=-\$10$, matching $190-200=-\$10$.

Common mistake avoided. The first negative increment occurs from 1 to 2, but that does not prove output 1 maximizes profit. The later increase from 2 to 3 adds $6, making output 3 the best of the five choices.

Unchanged Costs and the Scope of the Decision

Suppose total cost consists of a fixed amount $F$, which is identical at two output levels, plus a cost that varies with output. In the difference in total cost, the fixed amount cancels:

\Delta TC=[F+C_{\mathrm{variable}}(q+1)]-[F+C_{\mathrm{variable}}(q)] =C_{\mathrm{variable}}(q+1)-C_{\mathrm{variable}}(q).

This is why an unchanged fixed cost does not alter the incremental comparison. It still belongs in total profit, and it can matter for other decisions. The cancellation applies only when the fixed cost is the same at both output levels. If producing the additional unit requires a new shift, an extra equipment rental, or another step-up in fixed expense, that added expense must be included in the cost change.

The test also depends on the size of the proposed increment. A move from output 2 to output 5 is not automatically the same as evaluating three separate one-unit changes. The total change across the move is the sum of the changes over the intervening steps, and one or more intermediate increments could reduce profit. State clearly whether a problem asks about one more unit or a larger change.

Common Traps and Practical Use

Comparing totals instead of changes. $TR(q+1)>TC(q+1)$ means profit at $q+1$ is positive; it does not establish that profit rose from $q$.
Reversing the subtraction. Define each change as ending value minus starting value. Then use $\Delta\pi=\Delta TR-\Delta TC$.
Stopping at the first negative increment. That shortcut is safe only when the sequence supports it. If incremental profit later becomes positive, compare total profits across the choices.
Dropping a relevant cost change. A cost that varies with the proposed output increase belongs in the comparison, whether it is called variable, fixed, or an expansion cost.
Treating a tie as a unique answer. If $\Delta\pi=0$, adjacent choices yield equal profit. Other feasible output levels may still be better or worse.

The same reasoning applies to a professional's hypothetical operating decision, such as whether to accept one additional service contract. The decision-maker would compare the extra revenue with every additional cost caused by accepting it, using the relevant time period and including any contract-specific expense. The labels and setting may change, but the decision rule remains a comparison of changes.

Key takeaway: An additional unit raises profit only when its added revenue exceeds its added cost. For a sequence of choices, use each incremental comparison to understand the changes, and compare total profits whenever the sequence does not provide a clear single turning point.

Check Your Understanding

For each problem, use consistent output levels and compare changes rather than relying only on total revenue or total cost.

A hypothetical furniture maker's revenue rises from $760 to $815 when output increases from 8 to 9 chairs. Its cost rises from $590 to $652. Find the profit change and state whether the ninth chair raises profit. (Answer: $\Delta\pi=\$55-\$62=-\$7$; the ninth chair lowers profit by $7.)
A repair firm has profit of $34 at its current workload. One additional job adds $125 in revenue and $91 in cost. Find profit after accepting the job and state whether the job is beneficial relative to the current workload. (Answer: Profit becomes $34+(125-91)=\$68$; the job raises profit by $34.)
A hypothetical grower has total revenue and cost of $240 and $190 at output 2, $310 and $270 at output 3, and $390 and $340 at output 4. Calculate both adjacent incremental profit changes and identify the best of these three outputs. (Answer: From 2 to 3, $\Delta\pi=70-80=-\$10$; from 3 to 4, $\Delta\pi=80-70=+\$10$. Profits are $50, $40, and $50, so outputs 2 and 4 tie for the highest profit.)
A small factory's total revenue increases by $96 from output 5 to 6, and total cost increases by $96. Profit at output 5 is $-\$18$. What is profit at output 6, and what can be concluded about these adjacent choices? (Answer: $\Delta\pi=96-96=\$0$, so profit at output 6 is also $-\$18$; the two choices tie.)
For a hypothetical publisher, profits at output levels 0 through 4 are $-\$20,-\$8,-\$13,-\$4,-\$9$, respectively. Calculate the adjacent profit changes and select the best listed output. (Answer: Changes are $+\$12,-\$5,+\$9,-\$5$; output 3 has the highest profit, $-\$4$.)

What You'll Learn