Shutdown Decisions With Cost Data

Why Cost Data Matter in a Shutdown Decision

A competitive firm can make an economic loss and still be better off producing than shutting down. The key is to compare the market price with the variable cost of production, not simply to ask whether revenue covers every cost. A cost table lets us make that comparison and identify the best available output without relying on a cost-curve graph.

The decision here is a short-run one: assume the firm can stop producing for the period, but must still pay its fixed cost. At any positive output, revenue may cover all variable cost and contribute something toward fixed cost, even if it does not cover total cost. If that contribution is positive, producing reduces the loss relative to shutting down. This tutorial is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Definition: Let $q$ be output, $P$ the market price per unit, $FC$ fixed cost, $VC(q)$ variable cost, and $TC(q)=FC+VC(q)$ total cost. For $q>0$, average variable cost is $AVC(q)=VC(q)/q$, and average total cost is $ATC(q)=TC(q)/q$. Profit from producing $q$ units is $\pi(q)=Pq-TC(q)$. If the firm shuts down and produces zero, its revenue and variable cost are zero, but it still pays $FC$, so its profit is $\pi(0)=-FC$.

For a given price, compare the profit from each feasible positive output with the shutdown profit. In a cost table, the most reliable method is to calculate total revenue, total cost, and profit at every listed output. The firm chooses the option with the greatest profit, including shutdown as an option. A negative profit means a loss; it does not, by itself, mean shutdown is best.

Short-run shutdown test: At a positive output $q$, operating is better than shutting down precisely when $Pq-VC(q)>0$. Since $VC(q)=q\,AVC(q)$, this is equivalent to $P>AVC(q)$. If price is above the minimum AVC across feasible positive outputs, at least one output covers its variable cost and contributes toward fixed cost. If price is below minimum AVC, every positive output has a negative contribution toward fixed cost, so shutdown is preferable. At equality, an output attaining minimum AVC gives zero contribution: operating at that output and shutting down yield the same profit.

Why the Test Uses Variable Cost

The shutdown comparison can be derived directly from profit. Operating at output $q>0$ earns $Pq-VC(q)-FC$. Shutting down earns $-FC$. Subtract the shutdown profit from the operating profit:

\pi(q)-\pi(0) =\bigl(Pq-VC(q)-FC\bigr)-(-FC) =Pq-VC(q) =q\bigl(P-AVC(q)\bigr).

The fixed cost cancels because it is paid under either choice. Since $q>0$, the sign of the difference depends on $P-AVC(q)$. This explains both why price is compared with average variable cost and why covering average total cost is not required to justify operating in the short run. The firm still accounts for fixed cost when calculating its total loss; fixed cost simply does not change the operating-versus-shutdown comparison.

Minimum AVC summarizes the test across outputs. When the price exceeds minimum AVC, some positive output gives a positive contribution $Pq-VC(q)$, so that output beats shutdown. The firm must still choose the best output: it should not assume that every output with price above its AVC is equally profitable. When price equals minimum AVC, operating is tied with shutdown only at an output attaining that minimum. Other positive outputs may have higher AVC and make the loss larger.

Worked Decisions From Cost Tables

The examples use discrete feasible quantities. For each one, total cost includes the fixed cost and the listed variable cost. The firm is a price taker, so it can sell each unit produced at the given market price. Because outputs are discrete, we compare total profit across the listed choices rather than treating output as continuously adjustable.

Worked Example: Operating at a Loss Can Be Better

A competitive workshop has a fixed cost of $30 for the period. Its variable costs at outputs of 0 through 4 units are shown below. If the market price is $11 per unit, determine the best decision and the resulting profit.

Output $q$	Variable cost $VC$	Average variable cost $AVC$	Total cost $TC$	Revenue at $P=11$	Profit
0	$0	—	$30	$0	−$30
1	$12	$12	$42	$11	−$31
2	$20	$10	$50	$22	−$28
3	$30	$10	$60	$33	−$27
4	$44	$11	$74	$44	−$30

Step 1: Find minimum AVC. At positive outputs, AVC values are $12, $10, $10, and $11. The minimum is $10. Since the price of $11 is above minimum AVC, at least one positive output can contribute toward fixed cost.

Step 2: Compare profits, including shutdown. For example, at $q=3$, total cost is $30+30=\$60$, revenue is $11\times3=\$33$, and profit is $33-60=-\$27$. The table shows that this is the highest profit among the listed choices. Shutdown would give $-FC=-\$30$.

Decision and check. Produce 3 units and accept a $27 loss. The variable-cost contribution at that output is $33-30=\$3$. Thus operating reduces the $30 shutdown loss by $3, matching the profit difference. The method is to compare total profit at each feasible output. The common mistake it avoids is to treat any operating loss as a reason to shut down.

Worked Example: Price Below Minimum AVC

A small processor has fixed cost of $24 and variable costs of $9, $16, $27, and $44 at outputs 1, 2, 3, and 4. The market price is $7 per unit. Decide whether it should produce and calculate the best profit.

Step 1: Calculate AVC and its minimum. The average variable costs are $9/1=\$9$, $16/2=\$8$, $27/3=\$9$, and $44/4=\$11$. Minimum AVC is $8, which is greater than the $7 price.

Step 2: Compare feasible profits. At $q=1$, total cost is $24+9=\$33$, revenue is $7, and profit is $7-33=-\$26$. At $q=2$, total cost is $24+16=\$40$, revenue is $14, and profit is $-\$26$. At $q=3$, profit is $21-(24+27)=-\$30$. At $q=4$, profit is $28-(24+44)=-\$40$. Shutdown gives $-\$24$.

Decision and check. Shut down; the loss is $24, less than the loss from any positive output. At $q=2$, for instance, revenue of $14 does not cover variable cost of $16, so producing also increases the loss by $2 relative to shutdown. The method is to compare the market price with minimum AVC and confirm against the feasible profits. The common mistake it avoids is choosing an output just because its revenue is positive.

Worked Example: Price Exactly Equals Minimum AVC

A packaging firm pays fixed cost of $40. Its variable costs at outputs 1 through 4 are $15, $24, $33, and $48. The price is $11 per unit. Find minimum AVC and determine whether the firm operates, shuts down, or is indifferent between the two.

Step 1: Find AVC. At $q=1$, AVC is $15/1=\$15$; at $q=2$, it is $24/2=\$12$; at $q=3$, it is $33/3=\$11$; and at $q=4$, it is $48/4=\$12$. Minimum AVC is $11, exactly the market price.

Step 2: Calculate profit at the minimum-AVC output. At $q=3$, revenue is $11\times3=\$33$, total cost is $40+33=\$73$, and profit is $33-73=-\$40$. If the firm shuts down, profit is $-\$40$ as well.

Step 3: Check the other choices. At $q=1$, profit is $11-(40+15)=-\$44$. At $q=2$, it is $22-(40+24)=-\$42$. At $q=4$, it is $44-(40+48)=-\$44$. Thus neither another positive output nor shutdown gives a higher profit than producing 3 units.

Decision and check. The firm is indifferent between producing 3 units and shutting down: each yields a $40 loss. At $q=3$, revenue exactly covers variable cost, leaving no contribution toward fixed cost. The method is to identify the output attaining minimum AVC and then compare its profit with shutdown. The common mistake it avoids is claiming that equality requires the firm to produce: shutdown is equally good in this case.

Common Trap: Comparing Price With Average Total Cost

Suppose a candidate looks at the first example’s $q=3$ choice and calculates $ATC=60/3=\$20$. A mistaken conclusion is: “Price is $11, below ATC of $20, so the firm should shut down.” That comparison does show that producing 3 units does not cover total cost, but it does not show that shutdown is better.

The relevant comparison for whether this output beats shutdown is $P=\$11$ versus $AVC=\$10$. Revenue is $33 and variable cost is $30, so producing contributes $3 toward the $30 fixed cost. The operating loss is $30-3=\$27$, smaller than the $30 shutdown loss. Average total cost helps describe whether all costs are covered at an output; minimum average variable cost is the short-run threshold for whether operating can improve on shutdown.

Record the cost components.
Separate fixed cost from variable cost. Check that total cost at each positive output equals fixed cost plus variable cost.

Calculate AVC.
Divide variable cost by output for every positive feasible quantity. Do not calculate AVC at zero output.

Use minimum AVC as a quick test.
If price is below minimum AVC, shutdown beats every positive output. If price equals or exceeds minimum AVC, compare operating choices with shutdown, taking account of whether equality or a positive contribution applies.

Choose the highest-profit option.
For each feasible output, calculate revenue as $Pq$, total cost as $FC+VC$, and profit as revenue minus total cost. Include shutdown profit of $-FC$.

What the Decision Does—and Does Not—Say

A shutdown calculation answers a specific short-run question: given the period’s price and costs, does producing beat producing zero? A firm that operates may still report an economic loss because its contribution toward fixed cost is not large enough to cover the entire fixed cost. Conversely, a positive price does not guarantee that producing is worthwhile: if price is below minimum AVC, each feasible positive output generates too little revenue to cover its variable cost.

Cost tables can have several outputs with the same AVC or profit. Calculate the listed values rather than inferring a unique choice from a comparison alone. In particular, when price equals minimum AVC, an output attaining that minimum ties with shutdown; if several outputs attain the minimum, each has the same contribution of zero. Always report the output and profit, not just the words “operate” or “shut down.”

Check Your Understanding

For each problem, include shutdown as an option and use the stated fixed and variable costs.

A firm has fixed cost $18 and variable costs $8, $14, and $24 at outputs 1, 2, and 3. Price is $6. Should it operate or shut down, and what is its best profit? (Answer: Shut down; profit is −$18.)
A firm has fixed cost $50 and variable costs $14, $24, $39, and $60 at outputs 1 through 4. Price is $15. Which output maximizes profit, and what is that profit? (Answer: Either 2 or 3 units; profit is −$44.)
A firm has fixed cost $32 and variable costs $10, $18, $30, and $48 at outputs 1 through 4. Price is $9. Which positive output, if any, ties with shutdown, and what is the profit? (Answer: Output 2 ties with shutdown; profit is −$32.)
A firm has fixed cost $60 and variable costs $20, $34, $51, and $72 at outputs 1 through 4. Price is $16. What is minimum AVC, and what decision gives the highest profit? (Answer: Minimum AVC is $17; shut down for a profit of −$60.)
A firm has fixed cost $75 and variable costs $20, $34, $45, $64, and $90 at outputs 1 through 5. Price is $18. Find minimum AVC, the best output, and the resulting profit. (Answer: Minimum AVC is $15; produce 3 units; profit is −$66.)

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