Decisions At The Shutdown Price

What Happens at the Shutdown Price?

A firm deciding whether to produce compares the money it receives from selling output with the costs that producing adds. At the shutdown price, the price per unit is exactly the lowest average variable cost the firm can achieve. At that price, the firm can produce an output whose revenue just covers its variable cost. It does not, however, cover fixed cost.

This creates a tie in the short run: producing at a minimizing output and shutting down lead to the same loss, provided fixed costs must still be paid when production stops. The firm is not earning zero profit in either case. It is incurring a loss equal to fixed cost.

Definition: Let $q$ be output, $F$ be fixed cost, $V(q)$ be variable cost, and $p$ be the given price per unit. For $q>0$, average variable cost is $AVC(q)=V(q)/q$. The shutdown price is $p_s=\min_{q>0} AVC(q)$, where the minimum is taken over feasible positive output. A minimizing output is any positive output $q_s$ for which $AVC(q_s)=p_s$.

The short-run model used here assumes that the firm receives the given price for each unit sold, incurs total cost $F+V(q)$ when it produces $q$, and must pay $F$ even if it shuts down. Variable cost is zero at zero output, so shutdown means $q=0$, zero revenue, and a loss of $F$. The minimum average variable cost is assumed to be attained at a feasible positive output; if it is not attained, there is no output at which the equality case described here is achieved.

Why Producing and Shutting Down Tie

For positive output, total revenue is $pq$. Define the firm’s loss as total cost minus total revenue. Rewriting variable cost as $q\,AVC(q)$ shows how the price-to-cost comparison determines the loss:

\begin{aligned} L(q) &= F+V(q)-pq\\ &= F+q\,AVC(q)-pq\\ &= F+q\bigl(AVC(q)-p\bigr). \end{aligned}

At the shutdown price, $p=p_s$. By definition, $AVC(q)\geq p_s$ for every feasible positive output. Therefore $q(AVC(q)-p_s)\geq 0$, and producing cannot yield a loss smaller than $F$. At any output $q_s$ that minimizes average variable cost, the bracket equals zero, so the loss is exactly $F$. Shutting down also gives a loss of $F$, because the firm pays fixed cost but earns no revenue. That is the reason for the tie.

Theorem: If fixed cost is unavoidable in the short run and price equals the attained minimum average variable cost, then shutting down and producing at any output that minimizes average variable cost each result in a loss equal to fixed cost. Producing at any other feasible positive output gives a loss at least as large as fixed cost, and strictly larger if its average variable cost is above the minimum.

This result concerns the firm’s financial loss within the stated model. It does not say that production has no costs, or that the firm has recovered all of its costs. At a minimizing output, sales revenue covers variable cost exactly; fixed cost remains unpaid by revenue. If fixed costs could be avoided by shutting down, the comparison would have to use the costs that actually remain under shutdown, rather than assuming the same $F$ in both choices.

Worked Comparisons at Equality

Worked Example: Finding the Tied Output from a Cost Function

A hypothetical workshop has fixed cost $F=\$60$ per day and variable cost $V(q)=16+q^2$ dollars when it produces $q>0$ units per day. The price is $\$8$ per unit. Determine whether it should produce or shut down at this price, and calculate the loss under each choice.

Method. First find the minimum average variable cost, then compare revenue with variable cost at a minimizing output and account for fixed cost.

For positive output,

AVC(q)=\frac{16+q^2}{q}=\frac{16}{q}+q.

Differentiate to locate its minimum: $AVC'(q)=-16/q^2+1$. Setting this equal to zero gives $q^2=16$, so the positive candidate is $q=4$. Also, $AVC''(q)=32/q^3>0$ for $q>0$, confirming a minimum. Its value is $AVC(4)=16/4+4=8$. The given price, $\$8$, is therefore the shutdown price.

At $q=4$, variable cost is $V(4)=16+4^2=32$ dollars and revenue is $8\times4=32$ dollars. Total cost is $60+32=92$ dollars, so the loss is $92-32=\$60$. If the workshop shuts down, it earns no revenue and pays the $\$60$ fixed cost, giving a loss of $\$60$ as well.

As a check, producing $q=2$ would give variable cost $16+2^2=20$, revenue $8\times2=16$, and total cost $60+20=80$. Its loss would be $80-16=\$64$, which is greater than the $\$60$ loss from either shutdown or producing at $q=4$. The common mistake this calculation avoids is to ignore fixed cost after revenue covers variable cost.

Worked Example: Reading the Shutdown Price from a Schedule

A hypothetical producer has fixed cost $F=\$1{,}200$ per week. Its feasible production schedule is shown below. The variable-cost entries are total variable cost, not average variable cost. The product sells for $\$8$ per unit. Find the best choices at this price and compare their losses with shutting down.

Output $q$	Variable cost $V(q)$	Average variable cost $V(q)/q$
100	$1,200	$12
200	$1,600	$8
300	$2,700	$9

Method. Compute average variable cost at each positive feasible output and identify its minimum. The minimum is $\$8$ per unit, attained at $q=200$, so the given price is the shutdown price.

At $q=200$, revenue is $8\times200=\$1{,}600$, exactly equal to variable cost. Total cost is $1{,}200+1{,}600=\$2{,}800$. The loss is $2{,}800-1{,}600=\$1{,}200$. Shutting down also gives a loss of $\$1{,}200$.

Check the other listed outputs. At $q=100$, revenue is $\$800$, total cost is $1{,}200+1{,}200=\$2{,}400$, and loss is $\$1{,}600$. At $q=300$, revenue is $\$2{,}400$, total cost is $1{,}200+2{,}700=\$3{,}900$, and loss is $\$1{,}500$. Thus, among the listed options, producing 200 units and shutting down tie for the lowest loss. This method avoids comparing total variable costs alone: the relevant comparison at each output is revenue against variable cost.

Worked Example: When Two Outputs Minimize Average Variable Cost

A hypothetical repair-parts maker has fixed cost $F=\$500$ per shift. For positive output, its average variable cost is $AVC(q)=8+(q-3)^2(q-6)^2$ dollars per unit. The market price is $\$8$ per unit. Determine whether producing three units, six units, or shutting down gives the smallest loss. Also check the loss from producing four units.

Method. Since a product of squares cannot be negative, $(q-3)^2(q-6)^2\geq0$, so $AVC(q)\geq8$. Equality holds at both $q=3$ and $q=6$. Thus the minimum average variable cost is $\$8$, and both outputs minimize it.

At $q=3$, variable cost is $3\times8=\$24$ and revenue is $3\times8=\$24$. Total cost is $500+24=\$524$, so the loss is $524-24=\$500$. At $q=6$, variable cost is $6\times8=\$48$, revenue is $6\times8=\$48$, and total cost is $500+48=\$548$. Its loss is $548-48=\$500$. Shutting down also costs $\$500$ in unavoidable fixed cost and earns no revenue, so its loss is $\$500$.

At $q=4$, the squared-product term is $(4-3)^2(4-6)^2=1\times4=4$, giving average variable cost $8+4=\$12$. Variable cost is $4\times12=\$48$, revenue is $4\times8=\$32$, and total cost is $500+48=\$548$. The loss is $548-32=\$516$, greater than the tied $\$500$ loss. The calculation confirms that every output minimizing average variable cost ties with shutdown, even when there is more than one such output.

Common Mistakes at the Equality

The most important trap is confusing the amount left after variable cost with total profit. At a minimizing output and the shutdown price, $pq=V(q)$, so revenue minus variable cost is zero. But total profit also subtracts fixed cost:

\text{Profit}=pq-V(q)-F=0-F=-F.

For example, in the workshop calculation above, the incorrect calculation “revenue equals variable cost, so profit is zero” omits the $\$60$ fixed cost. The correction is $32-32-60=-\$60$, a loss of $\$60$. A related trap is to assume that shutdown eliminates fixed cost. Under the short-run assumption in this tutorial, it does not.

A useful way to keep the choices straight is to calculate the loss, not just the contribution from sales after variable cost. At equality, the contribution is zero at a minimizing output, and the fixed cost remains. When comparing with shutdown, include the same unavoidable fixed cost in both choices. The firm is financially indifferent within this model; other considerations, such as costs of restarting, would require additional information not included here.

Check Your Understanding

For each hypothetical case, find the minimizing output or outputs where needed, then compare the loss at the stated price with the loss from shutting down.

A firm has fixed cost $\$90$ and the following schedule: at output 50, variable cost is $\$600$; at 100, it is $\$900$; at 150, it is $\$1{,}500$. Price is $\$9$ per unit. Which listed output ties with shutdown, and what is the loss? (Answer: Output 100 ties with shutdown; each choice gives a loss of $\$90$.)
A producer has $F=\$240$, $V(q)=25+q^2$ for positive $q$, and price $\$10$. Find the output minimizing average variable cost and calculate the loss from producing there and from shutting down. (Answer: $q=5$; both choices give a loss of $\$240$.)
A shop has $F=\$700$ and $AVC(q)=6+(q-2)^2$. Price is $\$6$. Compare producing two units, producing four units, and shutting down. (Answer: Two units and shutdown tie at a loss of $\$700$; producing four units gives a loss of $\$716$.)
A producer has fixed cost $\$350$ and this schedule: output 2 has $AVC=\$11$, output 4 has $AVC=\$8$, and output 6 has $AVC=\$8$. Price is $\$8$. Which outputs tie with shutdown, and what is the common loss? (Answer: Outputs 4 and 6, and shutdown, each give a loss of $\$350$.)
A firm has $F=\$1{,}250$ and $V(q)=36+q^2$ for positive $q$. Price is $\$12$. Find the output that minimizes average variable cost, verify whether price is the shutdown price, and state the loss if fixed cost falls by $\$200$ while variable cost and price stay unchanged. (Answer: $q=6$, minimum $AVC=\$12$, so price is the shutdown price; after fixed cost falls to $\$1{,}050$, producing at $q=6$ and shutting down each give a loss of $\$1{,}050$.)

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