The Short-Run Shutdown Condition

Why Compare Revenue With Variable Cost?

A competitive firm can sometimes reduce its loss by continuing to produce, even when its total revenue does not cover all of its costs. The key short-run question is not whether the firm earns an economic profit. It is whether producing adds more to revenue than it adds to the costs that can be avoided by shutting down.

A shutdown means choosing zero output for the period. Under the usual short-run model, some costs—such as a lease already committed for the period—remain payable whether the firm produces or not. Those costs affect the size of the firm’s loss, but they do not decide whether producing is better than shutting down. The comparison is between revenue and variable cost.

Definition: Let $q\geq 0$ be the firm’s output, $P$ the market price per unit, $F$ the fixed cost that remains payable in the short run, and $VC(q)$ variable cost, with $VC(0)=0$. For $q>0$, average variable cost is $AVC(q)=VC(q)/q$. The shutdown price is the minimum value of average variable cost over feasible positive output levels, when that minimum is attained.

This rule assumes a price-taking firm: it can sell its output at the given market price, so revenue from producing $q$ units is $Pq$. It also assumes that the fixed cost in the comparison is unavoidable for the period and that there are no additional shutdown or restart costs. If a cost can be avoided by stopping production, it belongs in the avoidable-cost comparison rather than in the unavoidable fixed cost.

Deriving the Shutdown Condition

The firm compares its profit from producing a positive quantity with its profit from producing zero. When it shuts down, revenue is zero and it still pays $F$, so shutdown profit is $-F$. At a positive output $q$, profit is revenue less both fixed and variable cost.

\pi(q)=Pq-VC(q)-F,\qquad \pi(0)=-F.

Subtracting shutdown profit from operating profit isolates the effect of production. The fixed cost cancels because it is paid in either case. The difference is the contribution from operating: revenue less variable cost.

\pi(q)-\pi(0) =\bigl(Pq-VC(q)-F\bigr)-(-F) =Pq-VC(q).

For positive output, $VC(q)=q\,AVC(q)$. Therefore the contribution from operating can also be written as $q[P-AVC(q)]$. Because $q>0$, its sign depends on whether price is above or below average variable cost at that output. Producing is better than shutting down if there is a feasible positive output for which price exceeds its average variable cost.

Pq-VC(q)=q\bigl[P-AVC(q)\bigr].

If average variable cost has a minimum at a feasible positive output, this comparison gives the short-run shutdown rule. When price is below the minimum, every positive output has variable cost per unit above price, so producing adds to the firm’s loss. When price is above the minimum, at least one positive output has price above average variable cost, so operating improves on shutting down. At equality, the firm is indifferent between shutdown and producing at an output that minimizes average variable cost.

Theorem: Under the stated short-run assumptions, a competitive firm shuts down if $P<\min AVC$. It is better off operating if $P>\min AVC$. At $P=\min AVC$, shutdown and production at a minimizing output give the same profit. Thus “operate when price covers minimum average variable cost” includes equality as a tie, not as a strictly better outcome.

Why the Minimum of Average Variable Cost Matters

The minimum average variable cost is the lowest price at which any positive output can cover its variable cost. If price is below that level, choosing a different positive quantity cannot solve the problem: each feasible quantity has variable cost greater than revenue. If price is above it, the output that minimizes average variable cost provides a direct comparison showing that production can cover variable cost. The firm might choose a different output to maximize profit, but that further choice is not needed to establish whether shutting down is preferable.

For a differentiable variable-cost function, the relationship between marginal cost and average variable cost helps locate the minimum. Marginal cost is the additional variable cost from a small increase in output, so $MC(q)=dVC(q)/dq$. Differentiating average variable cost gives the following result:

\frac{d\,AVC(q)}{dq} =\frac{q\,MC(q)-VC(q)}{q^2} =\frac{MC(q)-AVC(q)}{q}.

At an interior minimum of average variable cost, its derivative is zero. Since $q>0$, the last expression is zero exactly when marginal cost equals average variable cost. This is a condition for locating a smooth interior minimum, not a replacement for checking that the point is a minimum and feasible.

\frac{d\,AVC(q)}{dq}=0 \quad\Longrightarrow\quad MC(q)=AVC(q).

The derivation follows from the quotient rule and the definition of marginal cost: $AVC(q)=VC(q)/q$, so its derivative is $[q\,VC'(q)-VC(q)]/q^2$. Substituting $VC'(q)=MC(q)$ and $VC(q)/q=AVC(q)$ yields the displayed expression. At an interior minimum, the derivative is zero; multiplying by the positive quantity $q$ gives $MC(q)=AVC(q)$.

Worked Example: A Smooth Variable-Cost Function

A price-taking workshop has short-run fixed cost $F=\$200$ and variable cost $VC(q)=q^3-6q^2+15q$, where $q>0$ is measured in batches and costs are in dollars. Determine the shutdown threshold and decide whether a price of $\$8$ per batch makes operation preferable to shutdown.

Method. Find the minimum average variable cost, then compare the price with that threshold. This tests the shutdown decision without requiring a full calculation of the profit-maximizing output.

First divide variable cost by output:

AVC(q)=\frac{q^3-6q^2+15q}{q} =q^2-6q+15 =(q-3)^2+6.

Because $(q-3)^2\geq 0$, the minimum average variable cost is $\$6$ per batch, attained at $q=3$. At the proposed price, $\$8>\$6$, so operation can improve on shutdown. To verify the comparison at the minimizing output, variable cost at $q=3$ is $27-54+45=\$18$, and revenue is $8(3)=\$24$. Operating therefore adds $\$24-\$18=\$6$ relative to shutdown.

The firm’s profit at that output is $\$24-\$18-\$200=-\$194$. Shutdown profit is $-\$200$. Thus the firm can be better off operating even though it still makes a loss. At the threshold price of $\$6$, revenue at $q=3$ would be $6(3)=\$18$, exactly equal to variable cost, so operating and shutting down would tie at a profit of $-\$200$.

Check. The contribution at $q=3$ and price $\$8$ is positive, and operating profit is $\$6$ greater than shutdown profit. The example avoids the mistake of treating any negative total profit as an automatic reason to shut down.

Reading Cost Data and Handling Equality

The same reasoning works when output choices are discrete rather than continuous. Calculate average variable cost at each feasible positive output and find the smallest value. Compare the market price with that value. At equality, production at the minimizing output does not increase profit over shutdown; it merely avoids making the loss any larger.

Worked Example: Discrete Output Choices

A small producer can make only 5, 10, or 15 units during the period. Its fixed cost is $\$300$, and its variable costs at those output levels are $\$60$, $\$100$, and $\$180$, respectively. At a market price of $\$11$ per unit, should it operate?

Method. Calculate average variable cost at every feasible positive output, then check the operating-profit comparison. This avoids dividing total cost, including fixed cost, by output.

At 5 units, $AVC=\$60/5=\$12$ per unit. At 10 units, $AVC=\$100/10=\$10$ per unit. At 15 units, $AVC=\$180/15=\$12$ per unit. The minimum is $\$10$, attained at 10 units. Since the price of $\$11$ is above this threshold, operating is preferable.

The comparison can be checked directly at each available output. At 5 units, revenue is $5(\$11)=\$55$, so revenue less variable cost is $\$55-\$60=-\$5$. At 10 units, it is $\$110-\$100=\$10$. At 15 units, it is $\$165-\$180=-\$15$. Shutdown gives a contribution of zero, so producing 10 units is better than shutdown among the listed choices.

Including fixed cost, profit at 10 units is $\$110-\$100-\$300=-\$290$, while shutdown profit is $-\$300$. At a price of $\$10$, revenue at 10 units equals variable cost, so operating and shutting down would both produce profit of $-\$300$. The example avoids using average total cost as the shutdown threshold.

Shutdown Is Not the Same as Break-Even

Break-even asks whether revenue covers total cost at a particular output. The shutdown decision asks whether producing is better than producing nothing in the short run. A firm can therefore rationally operate while making an economic loss: revenue may cover variable cost and contribute something toward unavoidable fixed cost without covering all of that fixed cost.

The distinction is visible in a cost table. Suppose the only feasible outputs are 20, 40, and 60 units. Fixed cost is $\$900$, and variable costs are $\$1{,}000$, $\$1{,}600$, and $\$3{,}000$, respectively. Average variable costs are $\$50$, $\$40$, and $\$50$. At price $\$45$, the firm can cover variable cost at 40 units, even though its revenue there is only $\$1{,}800$ and total cost is $\$2{,}500$. Its operating loss is $\$700$; shutdown would lose $\$900$. The firm prefers operation by $\$200$.

The shutdown rule is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus. Its practical value is that it focuses attention on the costs actually changed by the decision, rather than on the total accounting loss alone.

Common Mistakes

Comparing price with average total cost. Average total cost includes fixed cost. It is relevant to whether total cost is covered, but the short-run shutdown comparison is price versus average variable cost.
Shutting down whenever profit is negative. A loss can be smaller under operation if revenue exceeds variable cost and helps pay some unavoidable fixed cost.
Forgetting the equality case. At price equal to minimum average variable cost, production at a minimizing output and shutdown yield the same profit. Equality is a tie under this model.
Using total cost in place of variable cost. To compare operating with shutdown, subtract shutdown profit from operating profit. The unavoidable fixed cost cancels.

Key takeaway: For a competitive firm facing unavoidable short-run fixed cost, producing at $q>0$ changes profit relative to shutdown by $q[P-AVC(q)]$. The minimum average variable cost is the shutdown threshold: below it, shut down; above it, operate; at it, the firm is indifferent at a minimizing output.

Check Your Understanding

For each problem, compare price with the lowest average variable cost over the stated feasible outputs or output range.

A firm can produce 4, 8, or 12 units, with variable costs of $\$52$, $\$72$, and $\$132$. Fixed cost is $\$250$, and price is $\$10$ per unit. Find the minimum average variable cost and decide whether operation is preferable. (Answer: Minimum $AVC=\$9$ at 8 units; operate, with a $\$8$ contribution at 8 units.)
A firm has $VC(q)=q^3-6q^2+15q$ for $q>0$, fixed cost $\$80$, and price $\$5$ per unit. Find the minimum average variable cost and state the shutdown decision. (Answer: Minimum $AVC=\$6$ at $q=3$; shut down because price is below the threshold.)
A producer has feasible outputs of 10, 20, and 30 units, with variable costs of $\$140$, $\$200$, and $\$330$. Its fixed cost rises from $\$500$ to $\$800$, but remains payable if it shuts down. At price $\$11$ per unit, should it operate, and how does the fixed-cost change affect that decision? (Answer: Minimum $AVC=\$10$ at 20 units, so operate at either fixed-cost level; the higher fixed cost lowers both operating and shutdown profit by $\$300$, leaving their comparison unchanged.)
A firm’s minimum average variable cost is $\$18$, attained at a feasible positive output. Its unavoidable fixed cost is $\$700$, and price is $\$18$. Compare operating at the minimizing output with shutdown. (Answer: They tie; each gives profit $-\$700$.)
A firm can produce 5, 10, or 15 units, with variable costs of $\$75$, $\$100$, and $\$195$, respectively. Fixed cost is $\$400$, and price is $\$9$. Calculate the contribution at each output and choose between operation and shutdown. (Answer: Contributions are $-\$30$, $-\$10$, and $-\$60$; shut down, whose contribution is $\$0$.)

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