From a Schedule to a Decision
A cost-and-revenue schedule can show a firm’s best choice even when no graph or calculus is involved. The key is to separate two questions: which output gives the highest profit among production choices? And does producing at that output beat shutting down temporarily? These questions are related, but they are not identical.
This analysis applies to a short-run decision with a finite set of feasible output levels. It does not require assuming that the firm is a price taker: total revenue can be read directly from the schedule. To compare production with shutdown, however, we must know which costs continue if production stops. The model below assumes that fixed costs are unavoidable in the short run, while the listed variable costs can be avoided by shutting down.
At zero output, the firm has no sales and avoids variable production costs, but it still pays \(FC\). Therefore, the shutdown option is not automatically a profit of zero. If the highest profit from production is negative, compare that loss with \(-FC\): the less negative option is better. A firm can rationally produce while making an operating loss if production reduces the loss it would otherwise incur.
A Reliable Schedule Method
For every feasible output, calculate total cost and operating profit. Then compare the best producing result with shutdown. Keeping the shutdown row in the same table makes it harder to mistake the best operating quantity for the best overall decision.
Record each available output, including zero if shutdown is an option. Use the revenue and cost data for the same time period and units.
At each positive output, add fixed cost and variable cost: \(TC(q)=FC+VC(q)\).
Subtract total cost from total revenue: \(\pi(q)=TR(q)-TC(q)\). The largest value is the best production choice.
Calculate shutdown profit using costs that remain due. In this model, \(\pi_{SD}=-FC\). Choose shutdown only if this is greater than the best operating profit.
When a schedule reports total cost rather than variable cost, fixed cost can be separated by subtracting it from total cost at each positive output. In the stated model, the difference should be the avoidable variable cost. Check that the zero-output cost is consistent with the fixed cost that remains due under shutdown.
Why the Shutdown Test Works
The shutdown decision is a comparison of two profit levels, not a test of whether operating profit is positive. Subtract shutdown profit from profit at a positive output:
The fixed cost cancels because it is paid under either choice. Thus, producing at \(q\) is better than shutting down precisely when \(TR(q)>VC(q)\); the firm is indifferent when \(TR(q)=VC(q)\), and shutdown is better when \(TR(q)<VC(q)\). This comparison does not, by itself, identify the most profitable production quantity. The firm must still calculate profit at all feasible outputs, since revenue and variable cost can change differently as output changes.
For a positive output, the condition \(TR(q)\ge VC(q)\) can also be written as average revenue being at least average variable cost, where average revenue is \(TR(q)/q\) and average variable cost is \(VC(q)/q\). This version applies when \(q>0\); the direct comparison of total revenue and avoidable cost works without dividing by output. The model’s shutdown comparison must be adjusted if some costs listed as fixed can actually be avoided by stopping, or if shutdown itself creates an additional cost.
Worked Example: Find the Best Output and Compare It with Shutdown
A hypothetical workshop can produce 0, 1, 2, 3, or 4 batches in a short period. Its unavoidable fixed cost for the period is $40. The variable cost and total revenue for each feasible output are listed below. Determine the best production quantity and whether the workshop should operate.
| Output \(q\) | Variable cost \(VC(q)\) | Total revenue \(TR(q)\) |
|---|---|---|
| 0 | $0 | $0 |
| 1 | $25 | $35 |
| 2 | $48 | $68 |
| 3 | $78 | $90 |
| 4 | $116 | $100 |
Method. Compute total cost and profit at each positive output, identify the best operating profit, and compare it with shutdown. The shutdown cost is still $40, so shutdown profit is \(-\$40\).
At \(q=1\), total cost is \(40+25=\$65\), and operating profit is \(35-65=-\$30\).
At \(q=2\), total cost is \(40+48=\$88\), and operating profit is \(68-88=-\$20\).
At \(q=3\), total cost is \(40+78=\$118\), and operating profit is \(90-118=-\$28\).
At \(q=4\), total cost is \(40+116=\$156\), and operating profit is \(100-156=-\$56\). The operating profits are therefore \(-\$30\), \(-\$20\), \(-\$28\), and \(-\$56\). The best production choice is \(q=2\), with a loss of $20. Since \(-\$20>-\$40\), producing two batches is better than shutting down.
Check. At \(q=2\), revenue less variable cost is \(68-48=\$20\), the amount by which production improves on shutdown. The profit comparison agrees: \(-20-(-40)=\$20\). The common mistake of treating shutdown profit as zero would incorrectly lead the decision-maker to shut down.
Losses Do Not Automatically Mean Shutdown
A negative operating profit tells us that total revenue is less than total cost; it does not tell us whether shutdown is preferable. Fixed costs already committed for the short run may be incurred whether the firm operates or not. If revenue covers the avoidable costs of production, operating can offset part of those fixed costs.
Worked Example: A Loss-Making Firm That Should Shut Down
A hypothetical repair service owes $35 in unavoidable fixed costs for the week. Its variable cost can be avoided if it stops taking jobs. The schedule gives the following revenue and variable cost at each positive job volume. Find the best overall decision.
| Jobs \(q\) | Variable cost \(VC(q)\) | Total revenue \(TR(q)\) |
|---|---|---|
| 0 | $0 | $0 |
| 1 | $24 | $20 |
| 2 | $50 | $45 |
| 3 | $78 | $70 |
Method. Add the $35 fixed cost to each positive-output variable cost, then compare each operating profit with shutdown profit, \(-\$35\).
At \(q=1\), total cost is \(35+24=\$59\), so profit is \(20-59=-\$39\). At \(q=2\), total cost is \(35+50=\$85\), so profit is \(45-85=-\$40\). At \(q=3\), total cost is \(35+78=\$113\), so profit is \(70-113=-\$43\).
The best operating quantity is \(q=1\), with a loss of $39. But shutdown gives a smaller loss of $35, because \(-\$35>-\$39\). The repair service should shut down for the week.
Check. At one job, revenue less variable cost is \(20-24=-\$4\). Taking the job adds $4 of avoidable cost beyond the revenue it brings in, making the firm $4 worse off than shutdown. The result is consistent with the profit comparison: \(-39-(-35)=-\$4\).
Use the Full Schedule, Not Revenue Alone
The quantity with the greatest revenue need not have the greatest profit. Output can raise revenue while raising variable cost even more. Conversely, a quantity with a loss can be the best choice if it loses less than the alternatives and less than shutdown. The schedule must be evaluated row by row unless a separately established result allows the candidate choices to be narrowed.
For a discrete schedule, the change in operating profit between adjacent output levels is the change in revenue minus the change in variable cost. Fixed cost does not change between those operating choices. This incremental comparison helps explain why profit can fall even as revenue rises.
Worked Example: Revenue Peaks at a Different Output Than Profit
A hypothetical print shop has an unavoidable fixed cost of $12 thousand for a production period. It can choose any of four listed output levels. All revenue and variable-cost figures below are in thousands of dollars. Determine the best operating output, compare it with shutdown, and explain why the highest-revenue output is not best.
| Output \(q\) | Variable cost (thousands) | Total revenue (thousands) |
|---|---|---|
| 0 | 0 | 0 |
| 1 | 7 | 9 |
| 2 | 13 | 19 |
| 3 | 20 | 30 |
| 4 | 29 | 38 |
Method. Calculate operating profit as revenue minus variable cost minus fixed cost, keeping all amounts in thousands of dollars.
At \(q=1\), total cost is \(12+7=19\) thousand, so profit is \(9-19=-10\) thousand. At \(q=2\), total cost is \(12+13=25\) thousand, so profit is \(19-25=-6\) thousand. At \(q=3\), total cost is \(12+20=32\) thousand, so profit is \(30-32=-2\) thousand. At \(q=4\), total cost is \(12+29=41\) thousand, so profit is \(38-41=-3\) thousand.
The best operating output is \(q=3\), with a loss of $2 thousand. Shutdown profit is \(-12\) thousand, so operating at \(q=3\) is better. Although revenue is highest at \(q=4\), moving from \(q=3\) to \(q=4\) raises revenue by \(38-30=8\) thousand while raising variable cost by \(29-20=9\) thousand. Profit therefore falls by \(8-9=-1\) thousand, from \(-2\) thousand to \(-3\) thousand.
Check. At \(q=3\), revenue less variable cost is \(30-20=10\) thousand, which is greater than zero and therefore improves on shutdown. At \(q=4\), the corresponding amount is \(38-29=9\) thousand. Both outputs beat shutdown, but \(q=3\) gives the higher profit.
Common Exam Traps
A schedule can invite several different errors. The following checks keep the production decision and shutdown decision distinct.
- Assuming shutdown profit is zero. If fixed costs remain due, use \(-FC\) as shutdown profit. In the first example, the wrong comparison is \(-\$20<\$0\), which might suggest shutdown. The correct comparison is \(-\$20>-\$40\), so production at two batches is better.
- Choosing the output with highest revenue. Revenue does not subtract costs. In the print-shop example, \(q=4\) earns $38 thousand in revenue, but its $3 thousand loss is worse than the $2 thousand loss at \(q=3\).
- Shutting down whenever operating profit is negative. Compare the loss from operating with the loss from shutdown. A firm may operate at a loss if it covers its avoidable costs and thereby contributes toward fixed costs.
- Comparing revenue with total cost to decide whether to shut down. That comparison determines whether total operating profit is positive. For the shutdown comparison in this model, compare revenue with avoidable variable cost, because fixed cost is paid either way.
- Mixing units or periods. Revenue, variable cost, and fixed cost must refer to the same period and use the same units. A weekly revenue figure cannot be compared directly with an annual fixed cost without converting one of them.
Check Your Understanding
For each problem, calculate the best operating choice and compare it with shutdown. Unless stated otherwise, fixed costs remain due during shutdown, and the listed variable costs are avoidable.
- A hypothetical food cart has unavoidable fixed costs of $18 per day. For outputs of 1, 2, and 3 trays, variable costs are $12, $22, and $33, while revenues are $15, $28, and $36. Which output maximizes operating profit, and should the cart operate? (Answer: Produce 2 trays; operating profit is \(-\$12\), compared with shutdown profit of \(-\$18\).)
- A hypothetical mobile service owes $25 in fixed costs for the day. At outputs of 1, 2, and 3 appointments, variable costs are $18, $34, and $50, and revenues are $16, $30, and $45. What is the best overall choice? (Answer: Shut down; the best operating profit is \(-\$27\) at 1 appointment, while shutdown profit is \(-\$25\).)
- A hypothetical craft producer has fixed costs of $40. For outputs of 1, 2, and 3 units, variable costs are $20, $39, and $55, while revenues are $26, $50, and $69. Find the best operating quantity and state whether the producer should operate. (Answer: Produce 3 units; profit is \(-\$26\), which is better than shutdown profit of \(-\$40\).)
- A hypothetical studio has unavoidable fixed costs of $30. Its total costs at outputs 0, 1, 2, and 3 are $30, $50, $69, and $88. Revenues at those outputs are $0, $24, $46, and $63. Calculate variable cost at each positive output, then give the best overall decision. (Answer: Variable costs are $20, $39, and $58; produce 2 units, earning \(-\$23\), rather than shutting down at \(-\$30\).)
- A hypothetical workshop has fixed costs of $50. At outputs of 1, 2, 3, and 4 units, variable costs are $27, $49, $75, and $103, and revenues are $25, $53, $77, and $105. Find the best operating output and compare its profit with shutdown; do not choose by revenue alone. (Answer: Produce 2 units; its profit is \(-\$46\), better than shutdown profit of \(-\$50\), and better than the other operating profits.)