Monopoly Shutdown Decisions and Fixed Costs

Why a Monopolist May Operate at a Loss

A monopoly does not necessarily shut down whenever its total profit is negative. In the short run, some costs may continue even when production stops. The relevant question is whether producing can cover its avoidable variable costs and contribute something toward those unavoidable costs. This tutorial develops that decision for a single-product monopolist, alongside the separate choice of output.

The distinction is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus. Assume a monopolist faces a downward-sloping inverse demand curve $P(q)$, chooses a nonnegative output $q$, and has total cost $C(q)=F+VC(q)$. Here, $F$ is a fixed cost that cannot be avoided by shutting down during the period considered, and $VC(q)$ is variable cost that is avoided when production stops. Revenue is $R(q)=P(q)q$. The model treats output as continuously variable and assumes the firm can choose shutdown, $q=0$.

Definition: The firm’s operating contribution at output $q$ is revenue minus avoidable variable cost, $R(q)-VC(q)$. Its operating profit is $R(q)-VC(q)-F$. If it shuts down in the short run, it earns no revenue, avoids variable cost, and still pays $F$, so shutdown profit is $-F$.

The short-run shutdown decision compares the best operating contribution with zero. Fixed cost is present on both sides of the comparison: it is paid whether the firm operates or shuts down. Therefore, a positive contribution makes operating more profitable than shutting down, even if operating profit remains negative.

\begin{aligned} \pi(q)&=R(q)-VC(q)-F,\\ \pi_{\text{shutdown}}&=-F,\\ \pi(q)-\pi_{\text{shutdown}}&=R(q)-VC(q). \end{aligned}

Output Choice and the Shutdown Test

First find the output that gives the greatest contribution among feasible operating quantities. Since $F$ is constant with respect to output, maximizing $R(q)-VC(q)-F$ over positive output gives the same output as maximizing $R(q)-VC(q)$. For an interior solution with differentiable revenue and variable cost, the first-order condition is marginal revenue equal to marginal cost. This condition identifies a candidate; to establish the best choice, check that it is a maximum and compare it with relevant boundaries, including shutdown.

Output condition: At an interior operating output $q^*>0$, a necessary first-order condition is $MR(q^*)=MC(q^*)$, where $MR(q)=R'(q)$ and $MC(q)=VC'(q)$. If this condition has no feasible solution, or does not identify a maximum, compare feasible boundaries and other candidates directly.

Once the best positive operating output has been found, compare its revenue with its avoidable variable cost. If $R(q^*)>VC(q^*)$, operating beats shutdown. If $R(q^*)<VC(q^*)$, shutdown is better. Equality means the firm is indifferent between that operating choice and shutdown. If the firm’s best contribution is at $q=0$, it does not gain from positive production.

\max_{q\geq 0}\left[R(q)-VC(q)\right] \begin{cases} >0 & \text{a positive contribution makes operation preferable},\\ =0 & \text{shutdown is at least as good; a positive-output tie may occur},\\ <0 & \text{every positive output loses contribution relative to shutdown.} \end{cases}

For a positive output, the same comparison can be written using average variable cost, $AVC(q)=VC(q)/q$. Since $R(q)=P(q)q$, revenue covers variable cost exactly when $P(q)\geq AVC(q)$. This is a check at the monopolist’s chosen output, not a substitute for finding that output and comparing it with shutdown. Unlike a firm taking a fixed market price, the monopolist’s price varies with output.

Why Fixed Cost Does Not Change the Output Condition

The result follows from a direct comparison. Let the firm change output from $q$ to $q+\Delta q$, while remaining in the short run and continuing to owe the same fixed cost. The change in profit is the change in revenue minus the change in variable cost; $F$ cancels. For a small output change, the marginal effect is $MR-MC$. At an interior maximum, this marginal effect is zero, giving $MR=MC$. The same cancellation explains why fixed cost does not change the shutdown comparison: shutdown and operation both include the same $F$.

\begin{aligned} \pi(q+\Delta q)-\pi(q) &=\bigl[R(q+\Delta q)-R(q)\bigr]\\ &\quad-\bigl[VC(q+\Delta q)-VC(q)\bigr],\\ \frac{d\pi}{dq}&=R'(q)-VC'(q)=MR-MC. \end{aligned}

This reasoning relies on the short-run assumption that the fixed cost is unavoidable over the decision period. If shutting down would allow the firm to avoid a cost, that cost belongs in the shutdown comparison rather than being treated as an unchanged fixed cost. The classification depends on what the firm can actually avoid in the relevant time period.

Worked Applications

Worked Example: Positive Profit at the Best Operating Output

A hypothetical monopolist faces inverse demand $P(q)=100-2q$ and has variable cost $VC(q)=10q+q^2$ and unavoidable short-run fixed cost $F=\$200$. Find its best operating output, price, and profit, then decide whether it should shut down. Output is continuous, and all prices and costs are in dollars.

Method. Find the interior contribution-maximizing output using $MR=MC$, check that it is a maximum, and compare its contribution with zero. The fixed cost is included when reporting profit but not when choosing output.

Revenue is $R(q)=(100-2q)q=100q-2q^2$, so $MR(q)=100-4q$. Variable cost gives $MC(q)=10+2q$. Equating them:

100-4q=10+2q \quad\Longrightarrow\quad 90=6q \quad\Longrightarrow\quad q^*=15.

The second derivative of contribution is $R''(q)-VC''(q)=-4-2=-6<0$, so this interior candidate is a maximum. The monopoly price is $P(15)=100-2(15)=\$70$. Revenue is $R(15)=70(15)=\$1{,}050$. Variable cost is $VC(15)=10(15)+15^2=150+225=\$375$. Thus, contribution is $\$1{,}050-\$375=\$675$, and operating profit is $\$675-\$200=\$475$. Shutdown profit is $-\$200$, so the firm should operate.

Check. Operating profit exceeds shutdown profit by $\$475-(-\$200)=\$675$, exactly the operating contribution. A common mistake would be to subtract the fixed cost in the $MR=MC$ equation; doing so would incorrectly make a cost that is unchanged with output affect the output choice.

Worked Example: Operating Despite a Total Loss

A hypothetical monopoly faces $P(q)=80-q$, variable cost $VC(q)=8q+2q^2$, and unavoidable short-run fixed cost $F=\$500$. Determine its best operating output and whether operation or shutdown gives the higher profit. Output is continuous, with prices and costs in dollars.

Method. Maximize contribution first, then compare operating profit with shutdown profit. Do not use the sign of total profit alone as the shutdown test.

Revenue is $R(q)=(80-q)q=80q-q^2$, so $MR(q)=80-2q$. Marginal cost is $MC(q)=8+4q$. The interior condition gives:

80-2q=8+4q \quad\Longrightarrow\quad 72=6q \quad\Longrightarrow\quad q^*=12.

Contribution has second derivative $-2-4=-6<0$, confirming a maximum. At $q=12$, price is $P(12)=80-12=\$68$, and revenue is $68(12)=\$816$. Variable cost is $8(12)+2(12^2)=96+288=\$384$. Contribution is $\$816-\$384=\$432$. Operating profit is $\$432-\$500=-\$68$, while shutdown profit is $-\$500$. Operating is better by $\$432$, so the firm should operate even though its total profit is negative.

As a check, $AVC(12)=384/12=\$32$, which is below the price of $\$68$; revenue therefore covers variable cost at the chosen output. A common mistake is to see the $\$68$ operating loss and conclude that shutdown is better. That ignores the $\$432$ contribution that helps pay the fixed cost.

Worked Example: No Positive Output Covers Variable Cost

A hypothetical monopolist faces $P(q)=18-q$, variable cost $VC(q)=20q+0.5q^2$, and unavoidable short-run fixed cost $F=\$90$. Decide whether it should produce, and show why a positive-output marginal condition is not enough when there is no feasible interior solution. Output is continuous, with prices and costs in dollars.

Method. Examine marginal contribution over feasible positive outputs and compare with the shutdown option. Do not force an interior $MR=MC$ solution if it implies infeasible output.

Revenue is $R(q)=18q-q^2$, so $MR(q)=18-2q$. Marginal cost is $MC(q)=20+q$. Equating them yields:

18-2q=20+q \quad\Longrightarrow\quad q=-\frac{2}{3}.

This output is infeasible because $q\geq0$. More directly, for every $q\geq0$, $MR-MC=(18-2q)-(20+q)=-2-3q<0$. Contribution therefore falls as positive output increases from zero, so no positive output improves on shutdown. For example, at $q=1$, price is $\$17$, revenue is $\$17$, variable cost is $20+0.5=\$20.50$, and contribution is $-\$3.50$.

At shutdown, the firm pays the unavoidable $\$90$, so shutdown profit is $-\$90$. The best choice is $q=0$, with profit $-\$90$. A common mistake is to treat the negative algebraic solution as a usable output or to assume every monopoly problem has a positive interior optimum. Feasibility and the shutdown option must also be checked.

Reading the Decision Correctly

The output problem and the shutdown problem answer different questions. The marginal condition identifies the best output conditional on operating at an interior solution. The shutdown comparison then asks whether the contribution at that output is positive relative to producing nothing. Fixed cost affects the profit level, but it does not alter either comparison when it is unavoidable in the short run.

Quantity being compared	Operating	Shutdown
Revenue	$R(q)$	0
Avoidable variable cost	$VC(q)$	0
Unavoidable fixed cost	$F$	$F$
Profit	$R(q)-VC(q)-F$	$-F$

A careful solution should show the candidate output, establish that it is feasible and profit-maximizing among operating choices, calculate revenue and variable cost, and compare the resulting profit with shutdown profit. If the operating contribution is positive but smaller than $F$, operation still minimizes the loss. If it is zero, shutdown and that operating choice yield the same profit. If it is negative, the firm does better by shutting down.

Key takeaway: In the short run, choose output by comparing marginal revenue and marginal cost, then compare the best operating contribution $R(q)-VC(q)$ with zero. An unavoidable fixed cost lowers profit under both operation and shutdown, so it does not by itself change the output or shutdown decision.

Check Your Understanding

For each hypothetical monopoly, use the short-run assumptions stated in the question. Find the best feasible choice and compare operating profit with shutdown profit.

A firm faces $P(q)=50-q$, $VC(q)=5q+q^2$, and unavoidable $F=\$180$. Find the interior output, price, operating profit, and shutdown decision. (Answer: $q=11.25$, price $\$38.75$, operating profit $\$73.13$, operate.)
A firm faces $P(q)=42-q$, $VC(q)=6q+q^2$, and unavoidable $F=\$200$. Find the best output and both profits. (Answer: $q=9$, operating profit $-\$38$, shutdown profit $-\$200$, operate.)
A firm faces $P(q)=25-2q$, $VC(q)=9q+q^2$, and unavoidable $F=\$40$. Find the best output and determine whether a total loss calls for shutdown. (Answer: $q=\frac{8}{3}$, price $\frac{\$59}{3}$, operating profit $-\frac{\$56}{3}$, shutdown profit $-\$40$, operate.)
A firm faces $P(q)=15-q$, $VC(q)=18q+q^2$, and unavoidable $F=\$25$. Check whether any positive output improves on shutdown. (Answer: $MR-MC=-3-4q<0$ for $q\geq0$, so shut down; profit is $-\$25$.)
A firm faces $P(q)=70-q$ and $VC(q)=10q+q^2$. Calculate the best output and contribution, then compare the decision under $F=\$300$ and $F=\$500$, with each fixed cost unavoidable during the period. (Answer: $q=15$, contribution $\$450$; operating profits are $\$150$ and $-\$50$, respectively, while shutdown profits are $-\$300$ and $-\$500$; operate under both fixed-cost values.)

Quantity being compared	Operating	Shutdown
Revenue	\(R(q)\)	0
Avoidable variable cost	\(VC(q)\)	0
Unavoidable fixed cost	\(F\)	\(F\)
Profit	\(R(q)-VC(q)-F\)	\(-F\)

What You'll Learn