Conditions for Long-Run Competitive Equilibrium

Why Long-Run Equilibrium Has More Than One Condition

A competitive firm takes the market price as given. In the short run, that price guides its output choice, but it does not by itself tell us whether firms will stay in the industry or whether new firms will enter. Long-run equilibrium adds those entry-and-exit considerations: firms must be choosing their best output, earning no positive or negative economic profit, and collectively supplying the quantity buyers want at the market price.

The key implication in the standard model is that the market price equals both marginal cost and the minimum average total cost of an active firm. This is a joint result of profit maximization, free entry and exit, and the assumed cost structure; it is not a rule that can be applied to every firm or market without checking those assumptions. This treatment is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus.

Model assumptions: The market is perfectly competitive, so each firm takes the market price $P$ as given. Firms produce a homogeneous product, have identical cost functions, and can enter or exit freely in the long run. Firms choose output to maximize economic profit. Assume an active firm has a smooth, U-shaped average total cost curve with a unique minimum, and that marginal cost intersects average total cost at that minimum. The market price adjusts so quantity demanded equals total quantity supplied.

The Conditions for Long-Run Competitive Equilibrium

Let $q$ be the output of one firm, $C(q)$ its total economic cost, and $P$ the market price per unit. Average total cost is $ATC(q)=C(q)/q$ for $q>0$. Marginal cost, $MC(q)$, is the additional cost of producing a small additional amount of output; with a differentiable cost function, it is $C'(q)$. A firm’s economic profit is revenue less all economic costs, including the opportunity cost of its resources.

Long-run competitive equilibrium: In the standard identical-firm model with free entry and exit, each active firm produces at the minimum of average total cost. At that output, $P=MC(q)=ATC(q)=\min_{z>0}ATC(z)$, so the firm’s economic profit is zero. In addition, the market clears: quantity demanded equals the total quantity supplied by active firms. No firm has an incentive to change its output, enter, or exit.

These statements describe related but distinct checks. The condition $P=MC(q)$ identifies the profit-maximizing output of an active price-taking firm when the firm has an interior optimum. Zero economic profit means the price just covers average total cost at that output. Free entry and exit make that zero-profit condition necessary for a long-run equilibrium in this model: persistent positive economic profit is inconsistent with no incentive for entry, while persistent economic loss is inconsistent with no incentive for exit. The combined conditions locate the active firm at minimum average total cost.

The market-clearing condition is separate. Even if every firm is producing at its efficient scale and earning zero economic profit, the market is not in equilibrium if buyers want a different total quantity from the one firms supply. If there are $N$ identical active firms, each producing $q$, total supply is $Nq$; market clearing requires quantity demanded at price $P$ to equal $Nq$.

Why Price Equals Minimum Average Total Cost

The connection follows from the profit equation and the shape of the cost curves. For an active firm, revenue is $Pq$, so economic profit can be written as $\pi(q)=Pq-C(q)$. An interior profit-maximizing choice by a price-taking firm satisfies $P=MC(q)$. In long-run equilibrium, free entry and exit require zero economic profit, which means $P=ATC(q)$. Together, these two conditions imply $MC(q)=ATC(q)$.

\pi(q)=Pq-C(q)=q\bigl(P-ATC(q)\bigr)

For a smooth cost function, differentiating average total cost shows why marginal cost equals average total cost at its minimum. Since $ATC(q)=C(q)/q$, the quotient rule gives the derivative below. At a smooth minimum, that derivative is zero. Under the assumed U-shaped cost structure, the equality occurs at the unique minimum of average total cost.

\frac{d\,ATC(q)}{dq} = \frac{qC'(q)-C(q)}{q^2} = \frac{MC(q)-ATC(q)}{q}

At the minimum, $d\,ATC(q)/dq=0$. Because $q>0$, the last expression can equal zero only if $MC(q)=ATC(q)$. With both the profit-maximizing condition $P=MC(q)$ and the zero-profit condition $P=ATC(q)$, the equilibrium price equals both costs at the efficient scale. This argument relies on an active firm, a differentiable cost function, and the stated cost-curve assumptions; it is not a claim that marginal cost equals average total cost at every output.

Worked Examples

Worked Example: Finding the Equilibrium Price and Firm Output

A hypothetical competitive industry has identical firms with long-run total cost $C(q)=q^2+16$, where cost is measured in dollars and output in units. Find the firm’s efficient-scale output and the long-run equilibrium price. Then verify the firm’s economic profit.

Method. Find average total cost, minimize it, and compare marginal cost at that output. This avoids assuming that a price-taking firm’s marginal-cost condition alone identifies the long-run equilibrium.

First, for $q>0$, average total cost is $ATC(q)=(q^2+16)/q=q+16/q$. Differentiate and set the derivative equal to zero: $d\,ATC/dq=1-16/q^2=0$. Thus $q^2=16$, and the positive output is $q=4$.

At this output, $ATC(4)=4+16/4=4+4=\$8$ per unit. Marginal cost is $MC(q)=C'(q)=2q$, so $MC(4)=2(4)=\$8$ per unit. Therefore the long-run equilibrium price is $P=\$8$ per unit.

Check economic profit directly. Revenue is $Pq=8(4)=\$32$. Total cost is $C(4)=4^2+16=16+16=\$32$. Hence $\pi=32-32=\$0$. Equivalently, $q(P-ATC)=4(8-8)=\$0$. The price, marginal cost, and minimum average total cost agree, and the firm breaks even in the economic sense.

Worked Example: Why a Marginal-Cost Output May Not Be a Long-Run Equilibrium

A hypothetical firm in a competitive industry has cost function $C(q)=q^2+9$. The market price is $P=\$7$ per unit. Determine its profit-maximizing output and economic profit, and assess whether this price satisfies the long-run equilibrium condition.

Method. First apply the firm’s price-taking output condition, then calculate total revenue and total cost. The trap to avoid is treating $P=MC$ as sufficient proof of long-run equilibrium.

Marginal cost is $MC(q)=2q$. Setting price equal to marginal cost gives $7=2q$, so the firm’s chosen output is $q=3.5$ units. Total revenue is $Pq=7(3.5)=\$24.50$. Total cost is $C(3.5)=(3.5)^2+9=12.25+9=\$21.25$. Economic profit is therefore $\$24.50-\$21.25=\$3.25$.

The same result follows from average total cost. At $q=3.5$, $ATC=21.25/3.5=\$6.071428\ldots$ per unit, so profit is $3.5(7-6.071428\ldots)=\$3.25$. The equality $P=MC$ holds at the firm’s output, but the price exceeds minimum average total cost. To find that minimum, $ATC(q)=q+9/q$, so $d\,ATC/dq=1-9/q^2=0$, giving $q=3$. At $q=3$, $ATC=3+3=\$6$ and $MC=2(3)=\$6$. Thus the long-run equilibrium price for this cost function is $\$6$, not $\$7$. Positive economic profit at $\$7$ fails the free-entry long-run condition.

Worked Example: Checking Firm Conditions and Market Clearing

Suppose a hypothetical market has 12 identical active firms, each with long-run cost function $C(q)=q^2+25$. At a price of $\$10$ per unit, each firm produces 5 units, and market demand is 60 units. Verify whether this configuration satisfies the long-run competitive equilibrium conditions.

Method. Check the firm’s cost-minimizing scale, the relationship between price and marginal cost, the firm’s economic profit, and market clearing. This avoids checking only the individual firm while overlooking the market total.

Average total cost is $ATC(q)=q+25/q$. Its derivative is $1-25/q^2$, which equals zero at the positive output $q=5$. Average total cost there is $ATC(5)=5+25/5=5+5=\$10$ per unit. Marginal cost is $MC(q)=2q$, so $MC(5)=\$10$. Thus $P=MC=ATC=\$10$ at minimum average total cost.

For each firm, revenue is $10(5)=\$50$, and total cost is $C(5)=5^2+25=25+25=\$50$. Economic profit is $\$50-\$50=\$0$. Total market supply is $12(5)=60$ units, equal to the stated market demand of 60 units at $\$10$. The firm-level conditions and market-clearing condition all hold, so the configuration satisfies the standard long-run equilibrium tests.

What the Conditions Do—and Do Not—Say

A firm earns zero economic profit when its revenue covers both explicit costs and the opportunity costs of the resources it uses. This is not the same as saying that the firm has no revenue, no accounting profit, or no return to its owners. The result says that the firm earns no more than the return available from its next-best alternative, given how economic costs are defined.

The equality with minimum average total cost also has a cost-efficiency interpretation. Under the model’s assumptions, an active firm produces at the output with the lowest average cost per unit. A price above that minimum would allow positive economic profit at the efficient scale; a price below it would leave the firm unable to cover its average total cost there. Free entry and exit are what make a zero-economic-profit outcome the long-run benchmark.

These conditions are not universal facts about every industry. Barriers to entry, product differentiation, firms with different costs, or a cost curve that does not have the assumed unique minimum can change the analysis. Nor does the firm-level cost condition replace market clearing: the quantity buyers demand must still match total quantity supplied. For the standard competitive model, keep the checks distinct:

Firm output: an active firm chooses output where price equals marginal cost.
Long-run participation: free entry and exit are consistent with zero economic profit.
Efficient scale: at the assumed unique minimum of average total cost, marginal cost equals average total cost.
Market equilibrium: total quantity supplied equals quantity demanded at the market price.

Common Exam Trap

A common error is to find an output where $P=MC$, then label the result a long-run equilibrium without checking average total cost or profit. In the second worked example, $P=MC$ at $q=3.5$, but revenue is $\$24.50$ and cost is $\$21.25$, so the firm earns $\$3.25$ in economic profit. That is not consistent with long-run equilibrium under free entry.

The correction is to check all relevant conditions. Calculate $ATC$ at the chosen output, compute $\pi=q(P-ATC)$, and compare the price with minimum average total cost. In the example, the firm’s minimum $ATC$ is $\$6$, while the price is $\$7$. Thus the firm’s output is profit-maximizing at that price, but the industry is not in long-run equilibrium.

Key takeaway: For an active firm in the standard long-run competitive equilibrium, $P=MC=\min ATC$, and economic profit is zero. The market must also clear. A price equal to marginal cost by itself is not enough to establish all of these conditions.

Check Your Understanding

For each problem, use the standard identical-firm, free-entry-and-exit model and show the firm-level and market-level checks where the data allow them.

A hypothetical firm has cost function $C(q)=q^2+36$. Find its efficient-scale output and the long-run equilibrium price. Verify marginal cost and average total cost at that output. (Answer: $q=6$, $P=MC=ATC_{\min}=\$12$ per unit, and economic profit is zero.)
A firm has cost function $C(q)=2q^2+8$ and faces a price of $\$9$ per unit. Find its price-taking output and economic profit. Then state the long-run equilibrium price implied by its cost function. (Answer: $q=2.25$, economic profit $=\$2.125$, and the long-run equilibrium price is $\$8$ per unit.)
A firm has cost function $C(q)=q^2+16$ and faces a price of $\$9$ per unit. Calculate its profit-maximizing output and economic profit, then compare the price with minimum average total cost. (Answer: $q=4.5$, economic profit $=\$4.25$, minimum $ATC=\$8$ per unit, so this is not a long-run equilibrium.)
An industry has 15 identical firms with cost function $C(q)=q^2+25$. At price $\$10$, each firm produces 5 units, and quantity demanded is 75 units. Check the long-run firm conditions and market clearing. (Answer: Each firm has $P=MC=ATC_{\min}=\$10$, earns zero economic profit, and total supply $15(5)=75$ equals demand; all stated equilibrium conditions hold.)
An industry has 20 identical firms with cost function $C(q)=q^2+16$. At price $\$8$, each firm produces 4 units, but quantity demanded is 90 units. Are the firm-level conditions satisfied, and is the market in equilibrium? (Answer: The firm-level conditions hold, with $P=MC=ATC_{\min}=\$8$ and zero economic profit, but total supply is $20(4)=80$, not 90, so the market-clearing condition fails.)

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