Integrated Competitive-Firm Problem Workshop

One Firm, Several Linked Decisions

A competitive-firm problem often combines several steps: identify the price the firm takes as given, determine the firm’s revenue, choose output, decide whether to operate, and then consider how entry or exit changes the market over time. The order matters. In particular, a marginal-cost calculation identifies a candidate output, but a firm must choose the output that gives the highest profit over all feasible alternatives.

A competitive firm is a price taker: it can sell its output at the market price but cannot raise that price by reducing or expanding its own production. This is different from the market, whose total demand and supply determine the market price. The firm’s demand is horizontal at that price, while market demand generally slopes downward.

Definition: For a price-taking firm selling $q$ units at market price $P$, total revenue is $TR(q)=Pq$. If total cost is $C(q)=F+VC(q)$, where $F$ is fixed cost and $VC(q)$ is variable cost, economic profit is $\pi(q)=Pq-C(q)$. In the short run, shutting down means producing zero and avoiding variable cost, but fixed cost remains payable.

From the Market Price to the Firm’s Choice

Because the firm takes $P$ as fixed, its total-revenue function is a straight line through the origin. The additional revenue from one more unit is the price itself. This gives the central marginal comparison, but it does not by itself settle whether an output is globally best.

TR(q)=Pq,\qquad MR(q)=\frac{dTR}{dq}=P,\qquad \pi(q)=Pq-C(q)

If costs are differentiable, an interior profit maximum must satisfy $P=MC(q)$, where $MC(q)=C'(q)$. This condition follows because $d\pi/dq=P-C'(q)$: profit rises where price exceeds marginal cost and falls where marginal cost exceeds price. The condition is necessary for an interior maximum, but may not be sufficient if marginal cost is not well behaved. A firm with non-convex costs can have multiple local maxima, so it must compare the profit at candidate outputs and at shutdown.

Important condition: If variable cost is convex over the feasible output range, marginal cost is nondecreasing. Then $Pq-VC(q)$ is concave in output, and a point satisfying $P=MC(q)$, when it exists, maximizes operating profit globally. The firm still compares that operating profit with zero from shutdown. Without a suitable cost structure, compare profits across all relevant feasible outputs; do not treat every rising marginal-cost intersection as the firm’s supply.

Worked Example: Price, Revenue, Output, and the Shutdown Comparison

A small price-taking producer sells in a market where the market price is $P=\$14$ per unit. Its short-run cost is $C(q)=100+2q+q^2$ for $q\geq 0$. The variable cost is convex. Find the profit-maximizing output, profit, and shutdown decision.

Method. First calculate the firm’s revenue and marginal cost, then use the convex-cost condition to find the best operating output. Finally compare operating profit with the profit from producing zero.

The firm’s demand is horizontal at $\$14$; the market demand curve is not the firm’s demand curve. Total revenue is $TR(q)=14q$, and $MC(q)=2+2q$. Equating price and marginal cost gives $14=2+2q$, so $q=6$.

At $q=6$, revenue is $TR=14(6)=\$84$. Variable cost is $VC=2(6)+6^2=12+36=\$48$, and total cost is $C=100+48=\$148$. Thus profit from operating is $\pi=84-148=-\$64$.

If the firm shuts down, it produces zero, earns no revenue, avoids variable cost, and still pays the fixed cost of $\$100$. Its profit is therefore $-\$100$. Since $-64>-100$, it operates at six units even though it incurs an accounting loss in this short-run decision. The result is checked by the convexity of $VC(q)=2q+q^2$, which makes the marginal-cost choice the global operating-profit maximum.

Common mistake avoided: A loss does not automatically imply shutdown. The relevant comparison is operating profit, including fixed cost, versus the shutdown loss from unavoidable fixed cost—not versus zero profit.

Shutdown, Supply, and the Cost-Curve Picture

For positive output, a firm compares revenue with variable cost to decide whether operating is better than shutting down. Define average variable cost as $AVC(q)=VC(q)/q$ for $q>0$. Operating at output $q$ is at least as good as shutdown exactly when $Pq-VC(q)\geq 0$, or equivalently when $P\geq AVC(q)$. Thus the lowest price at which any output can cover variable cost is the minimum of average variable cost, when that minimum exists.

In a standard cost-curve graph, the price line is horizontal. The candidate operating quantity is where that line meets marginal cost on the relevant increasing branch. The shutdown price is the minimum AVC: below it, no positive output covers variable cost. These familiar graphical rules rely on the cost curves having the appropriate shape. If marginal cost is non-convex or has discontinuities, the graph can contain several candidate intersections; compare the profit at those candidates and any relevant endpoints instead of selecting an intersection mechanically.

Theorem: When variable cost is convex over feasible output and the firm can shut down at $q=0$, its short-run choice is an output maximizing $Pq-VC(q)$, or zero if the maximum operating surplus is negative. Under the usual smooth, single-minimum cost-curve conditions, positive supply lies on the increasing marginal-cost branch at or above minimum AVC. For arbitrary cost curves, the firm’s supply must be found by global profit comparison; the simple branch rule need not apply.

Worked Example: A Non-Convex Cost Curve and a Misleading Intersection

A price-taking firm faces $P=\$10$ and can produce any $q\geq 0$. Its marginal variable cost is $8+4q$ for $0\leq q\leq 1$, $20$ for $1<q\leq 2$, $1$ for $2<q\leq 20$, and $30$ for $q>20$. Variable cost is continuous, with $VC(0)=0$. Find the globally best output, allowing shutdown.

Method. Compare operating surplus $Pq-VC(q)$ at the candidate local maxima and at relevant boundaries. This is necessary because marginal cost falls sharply after $q=2$, so it is not a globally increasing cost curve.

On the first interval, $VC(q)=\int_0^q(8+4x)\,dx=8q+2q^2$. The equation $MC=P$ gives $8+4q=10$, so $q=0.5$. At that output, $VC=8(0.5)+2(0.5)^2=4+0.5=4.5$, revenue is $10(0.5)=5$, and operating surplus is $5-4.5=\$0.50$. This is a local maximum: marginal profit changes from positive to negative at $q=0.5$.

At $q=1$, $VC=8+2=10$, so operating surplus is $10-10=0$. From $q=1$ to $q=2$, marginal cost is $20$, greater than price, and surplus falls. At $q=2$, $VC=10+20(1)=30$, revenue is $20$, and surplus is $-\$10$.

From $q=2$ to $q=20$, marginal cost is $1$, so each additional unit raises operating surplus by $10-1=\$9$. At $q=20$, $VC=30+1(18)=48$, revenue is $10(20)=200$, and operating surplus is $200-48=\$152$. Beyond 20, marginal cost is $30>\$10$, so surplus falls. At shutdown the operating surplus is zero. Comparing the candidates gives the global choice $q=20$, with operating surplus $\$152$.

At $q=20$, $AVC=48/20=\$2.40$, which is below the price. The intersection at $q=0.5$ is indeed a local maximum, but it is not the global maximum. Selecting it as supply would miss the much more profitable output at 20 units.

Common mistake avoided: The condition $P=MC$ alone does not establish the global optimum. Check the cost-curve shape and compare distinct candidate outputs whenever the marginal-cost branch is not globally increasing.

Entry and Exit Link Firm Decisions to the Long Run

In the long run, firms can enter or leave an industry. With free entry and exit, identical firms, and no lasting barriers to adjustment, positive economic profit encourages entry and losses encourage exit. Entry or exit changes the number of firms and therefore market supply. In a long-run competitive equilibrium with active firms, economic profit is zero: market price equals the minimum long-run average total cost, and each firm produces at the output associated with that minimum. This result assumes firms can adjust all inputs in the long run and that the model’s entry and cost conditions support adjustment.

The graph interpretation is useful: a price above minimum average total cost gives positive economic profit, which attracts entry; a price below minimum average total cost gives losses, encouraging exit. The adjustment changes industry supply until the long-run price is consistent with zero economic profit. This is a long-run result, not a claim that every short-run firm immediately earns zero profit.

Worked Example: A Demand Increase, Short-Run Profit, and Long-Run Entry

Identical competitive firms have short-run and long-run cost $C(q)=q^2+16$ for $q>0$. Initially, market demand is $Q_D=160-10P$, the market price is $\$8$, and there are 20 firms. Demand then shifts to $Q_D=200-10P$. Find the short-run outcome after the shift and the long-run outcome, assuming free entry and exit.

Method. Find the firm’s cost-minimizing output at each relevant price, use the fixed number of firms for short-run market supply, and then impose the long-run zero-economic-profit condition.

Here $VC=q^2$, $MC=2q$, $AVC=q$, and $ATC=q+16/q$. At the initial price $P=8$, the firm chooses $q=4$, because $8=2q$ gives $q=4$. Its total cost is $4^2+16=32$, and revenue is $8(4)=32$, so initial economic profit is zero. Market quantity is $160-10(8)=80$, consistent with $20(4)=80$.

After the demand shift, in the short run each firm still takes price as given and supplies $q=P/2$, since $MC=2q$ and variable cost is convex. With 20 firms, market supply is $Q_S=20(P/2)=10P$. Market equilibrium requires $200-10P=10P$, so $20P=200$, $P=\$10$, and $Q=10(10)=100$. Each firm produces $q=10/2=5$.

At $P=10$ and $q=5$, each firm earns revenue $10(5)=\$50$ and has total cost $5^2+16=41$. Economic profit is $50-41=\$9$ per firm. The positive profit encourages entry.

For the long-run price, minimize $ATC=q+16/q$. Its derivative is $1-16/q^2$, which is zero at $q=4$; the average-cost curve falls before 4 and rises after 4, so this is its minimum. At that output, $ATC=4+16/4=8$. Thus long-run price is $\$8$ and each firm again produces 4 units. At that price, shifted market demand is $200-10(8)=120$. The number of firms is $120/4=30$. Entry raises the firm count from 20 to 30; the long-run check is zero profit, since revenue and total cost are both $\$32$ per firm.

Common mistake avoided: Do not use the long-run firm count to calculate the immediate short-run equilibrium. In the short run, the number of firms is fixed; entry changes the number only during the long-run adjustment.

A Reliable Problem-Solving Sequence

When a question combines an individual firm with the market, keep the two levels separate. The market price is a number the competitive firm takes as given; the firm’s chosen output then contributes to market supply. A disciplined calculation also makes shutdown and entry conditions explicit rather than inferring them from whether profit happens to be positive.

Identify the firm’s price.
Use market equilibrium or a stated market price, then treat that price as fixed in the individual firm’s revenue calculation.

Find output candidates.
Use $P=MC$ for smooth interior candidates, but check whether the cost structure makes a candidate a global maximum. Include feasible boundaries where relevant.

Compare operation with shutdown.
Calculate $Pq-VC(q)$ for operating choices and compare its maximum with zero operating surplus at shutdown. Fixed cost is still paid in the short run.

Separate time horizons.
Use the fixed number of firms for short-run market supply. For long-run adjustment, apply the stated entry and exit assumptions and check zero economic profit for active firms.

Common Errors in Integrated Problems

Using market demand as the firm’s demand. A competitive firm faces a horizontal demand curve at the market price; it does not choose a price from the market demand curve.
Choosing an output from $P=MC$ without checking it. With non-convex variable cost, a marginal-cost intersection can be a local maximum rather than the global maximum.
Shutting down whenever profit is negative. In the short run, compare operating surplus with zero; fixed cost is payable either way.
Applying the short-run supply rule without its conditions. The standard rising-$MC$-above-minimum-$AVC$ result relies on a suitable cost structure. Arbitrary cost curves require a global comparison.
Assuming long-run adjustment occurs immediately. Short-run firm count is fixed; entry and exit alter the number of firms over time.

Key takeaway: A complete competitive-firm solution links the market price to firm revenue, checks the global profit-maximizing output, compares operation with shutdown using variable cost, and distinguishes short-run decisions from long-run entry or exit.

Check Your Understanding

Work each problem through the firm’s choice and the relevant time horizon before checking the answer.

A price-taking firm faces $P=\$12$ and has short-run cost $C(q)=36+3q+q^2$. Find its profit-maximizing output, operating profit, and shutdown decision. (Answer: $q=4.5$, operating profit $-\$15.75$, so it operates rather than shutting down and losing $\$36$.)
At price $\$10$, a firm can choose $q=0,2,4,$ or $6$, with variable costs respectively $\$0,\$18,\$28,$ and $\$54$. Fixed cost is $\$30$. Which output maximizes profit, and what is profit at that output? (Answer: $q=4$; revenue is $\$40$, total cost is $\$58$, and profit is $-\$18$, better than the $-\$30$ shutdown profit.)
A competitive firm has $C(q)=q^2+25$. Find the output minimizing average total cost and the associated long-run zero-profit price. At that price, market demand is $Q_D=150-5P$; if firms are identical, how many firms operate in long-run equilibrium? (Answer: $q=5$, minimum $ATC=P=\$10$, market quantity $100$, and $20$ firms.)
A firm has $C(q)=40+2q+q^2$ and faces $P=\$6$. Calculate its profit-maximizing positive output and compare operating with shutdown. (Answer: $q=2$; operating profit is $-\$36$, which is better than shutdown profit $-\$40$, so it operates.)
A price-taking firm has increasing convex variable cost and faces a price below its minimum average variable cost. What is its short-run output? If the price rises above minimum average variable cost, what additional condition supports using the increasing marginal-cost branch as its supply? (Answer: It shuts down at the lower price; the standard rising-$MC$ supply rule is supported when variable cost is convex over feasible output, with the chosen output compared against shutdown.)

What You'll Learn