Why Compare Two Losses?
A firm can make a loss while it produces, yet still be better off producing than shutting down. The key is to compare the loss under each available choice, using the costs that actually change between those choices. In the short run, some fixed costs may remain payable even when output is zero. If those costs are incurred whether the firm operates or shuts down, they do not by themselves decide which choice gives the smaller loss.
This comparison can be made without assuming that producing earns a positive profit. It asks a narrower question: does the revenue from producing cover at least the variable cost that production requires? The answer follows directly by writing out both losses.
This setup assumes a short-run choice between producing at a selected positive output and shutting down at zero output; the same fixed cost is payable under either choice; variable costs are zero at zero output; and there are no additional shutdown or restart costs in the comparison. The fixed cost is not necessarily avoidable just because output stops. If shutting down changes the fixed costs payable, that difference must be included explicitly.
Deriving the Production Condition
Write the loss from each choice, then select the choice with the smaller loss. Producing is at least as good as shutting down when its loss is no greater than the shutdown loss:
Subtract \(TFC\) from both sides. The comparison reduces to:
The fixed cost cancels because it is paid in both alternatives. Thus, the production condition is not that revenue must cover all costs. It is that revenue must cover the variable costs associated with producing. If revenue is below variable cost, producing adds more variable cost than revenue and makes the loss larger than shutting down.
The reason the condition works is that \(TR-TVC\) measures the amount of fixed cost covered by operating revenue after paying variable costs. This amount is often called the contribution toward fixed cost. If it is positive, it offsets some of the fixed cost that remains payable either way. If it is negative, production fails to cover its variable cost and increases the loss.
Worked Comparisons
Worked Example: Producing Reduces the Loss
A workshop is considering producing 80 units for a short period. At that output, total revenue is $1,120, total variable cost is $960, and fixed cost is $500. The $500 fixed cost is payable whether the workshop operates or shuts down. Compare the two choices.
Method. Calculate the operating loss and shutdown loss separately. Do not decide from the sign of profit alone.
1. Operating loss. Total cost while producing is:
Revenue is $1,120, so the loss from producing is:
2. Shutdown loss. At zero output, variable cost is zero, but the same $500 fixed cost remains payable. The shutdown loss is therefore $500.
3. Compare. Producing loses $340, while shutting down loses $500. Producing reduces the loss by \(\$500-\$340=\$160\). The condition gives the same result: \(TR-TVC=\$1{,}120-\$960=\$160>0\), so production covers $160 of the fixed cost.
Check. The calculated operating loss is $160 less than the shutdown loss, matching the contribution \(TR-TVC\). The common mistake avoided here is to shut down just because total cost exceeds revenue; the relevant comparison is between the two losses.
Worked Example: Shutting Down Limits the Loss
A small mill is considering an order that would generate $900 of revenue and require $1,050 of variable cost. Its fixed cost is $300, and that cost remains payable if it shuts down. Determine which choice has the smaller loss.
1. Calculate the loss from producing. Total cost is:
Subtract revenue from total cost:
2. Calculate the loss from shutdown. The mill has no variable cost at zero output, but still pays its $300 fixed cost. Its shutdown loss is $300.
3. Compare. The operating loss of $450 is greater than the shutdown loss of $300, so shutting down limits the loss by $150. Equivalently, \(TR-TVC=\$900-\$1{,}050=-\$150\), meaning production adds $150 of cost beyond the revenue it earns.
Check. The difference between the operating and shutdown losses is \(\$450-\$300=\$150\), exactly the amount by which variable cost exceeds revenue. The common mistake avoided is treating the fixed cost as a reason to keep operating: in this comparison, it is paid either way.
Worked Example: Comparing Several Positive Output Levels With Shutdown
A seasonal packaging firm has a fixed cost of $400 that remains payable at zero output. Its revenue and variable cost at three possible production levels are shown below. Find the best choice among these output levels and shutdown, using only the loss comparison.
| Output | Total revenue | Total variable cost |
|---|---|---|
| 20 units | $300 | $250 |
| 40 units | $700 | $620 |
| 60 units | $960 | $900 |
1. Find the contribution at each positive output. Subtract variable cost from revenue:
2. Find the operating loss at each output. Since \(TFC=\$400\), operating loss is \(TFC-(TR-TVC)\):
3. Include shutdown. At zero output, the firm pays the $400 fixed cost and no variable cost, so its shutdown loss is $400.
4. Choose the smallest loss. The losses are $350, $320, $340, and $400 for outputs of 20, 40, and 60 units and shutdown, respectively. The smallest is $320, at 40 units. Each positive output covers its variable cost, so each beats shutdown; among them, 40 units has the greatest contribution and smallest loss.
Check. The best operating contribution is $80, so the associated loss is \(\$400-\$80=\$320\). This matches the direct comparison. The common mistake avoided is to stop after deciding that some production beats shutdown; when several outputs are available, compare their resulting losses as well.
When Fixed Costs Differ Between the Choices
The cancellation above depends on fixed cost being the same under production and shutdown. If shutdown allows some fixed cost to be avoided, the comparison must reflect that saving. Let \(TFC_P\) denote fixed cost when producing and \(TFC_S\) fixed cost when shut down. Under the same assumption of zero variable cost at shutdown, the losses are \(TFC_P+TVC-TR\) and \(TFC_S\). Therefore, producing is preferred when:
The right-hand side is the fixed cost saved by shutting down. Production must generate enough contribution to cover that amount. The familiar condition \(TR\geq TVC\) follows when \(TFC_P=TFC_S\), so that the fixed-cost saving is zero. This adjustment should be used only when the problem states that some fixed cost actually changes with the shutdown choice; a cost being called “fixed” does not, by itself, establish whether it is avoidable.
Worked Example: Accounting for Fixed Cost Avoided on Shutdown
A production run would generate $1,200 of revenue and incur $1,050 of variable cost. Fixed costs total $700 while producing, but only $500 remain payable if the firm shuts down. Compare the losses.
1. Loss from producing. Total cost is \(\$700+\$1{,}050=\$1{,}750\). The operating loss is \(\$1{,}750-\$1{,}200=\$550\).
2. Loss from shutdown. The firm still pays $500, so shutdown loss is $500.
3. Compare. Shutdown has the smaller loss by \(\$550-\$500=\$50\). The adjusted condition confirms this: contribution is \(\$1{,}200-\$1{,}050=\$150\), while the fixed-cost saving from shutting down is \(\$700-\$500=\$200\). Since \(\$150<\$200\), production does not cover the fixed cost that it could avoid.
Check. Directly comparing the losses and comparing contribution with the avoidable fixed-cost saving both recommend shutdown. The common mistake avoided is applying \(TR\geq TVC\) unchanged when fixed costs differ between the choices.
Common Mistake: Comparing Revenue With Total Cost Alone
A firm that has \(TR<TC\) is making an operating loss, but that fact alone does not determine whether it should shut down. In the first example, \(TR=\$1{,}120\) and \(TC=\$1{,}460\), so revenue is below total cost. The incorrect conclusion would be: “Revenue does not cover total cost, so shut down.” That calculation ignores the $500 fixed cost that is paid even after shutdown.
The correction is to calculate both alternatives. Producing loses \(\$1{,}460-\$1{,}120=\$340\); shutting down loses $500. Production is better by $160 because its revenue covers all $960 of variable cost and contributes $160 toward fixed cost. Always compare like with like: operating loss against shutdown loss, or contribution against the fixed cost that shutdown would actually save.
Check Your Understanding
For each problem, compare the loss from production with the loss from shutdown. Assume variable cost is zero at zero output unless the problem states otherwise.
- A print shop can produce 40 custom folders for revenue of $720. Variable cost is $680, and fixed cost of $240 remains payable if the shop shuts down. Which choice has the smaller loss? (Answer: Produce; operating loss is $200 versus a shutdown loss of $240.)
- A repair facility considers a job that yields $825 of revenue and requires $900 of variable cost. Fixed cost of $320 is payable under either choice. Calculate both losses and choose. (Answer: Shut down; operating loss is $395 versus a shutdown loss of $320.)
- A short production run has revenue of $1,450, variable cost of $1,300, and fixed cost of $600 payable under either choice. How much smaller is the preferred loss? (Answer: Produce; the operating loss is $450, which is $150 less than the $600 shutdown loss.)
- A maker of reusable containers has fixed cost of $400 whether it operates or shuts down. At outputs of 20, 40, and 60 units, \((TR,TVC)\) are respectively \((\$300,\$250)\), \((\$700,\$620)\), and \((\$960,\$900)\). Include shutdown as an option and identify the choice with the smallest loss. (Answer: Produce 40 units; its loss is $320, compared with losses of $350 at 20 units, $340 at 60 units, and $400 from shutdown.)
- A test-batch order yields $1,200 of revenue and costs $1,050 in variable cost. Fixed cost is $700 if producing but falls to $500 if shut down. Which choice is better, and by how much? (Answer: Shut down; the shutdown loss is $500, which is $50 less than the $550 operating loss.)