Long-Run Market Supply in a Constant-Cost Industry

Why Long-Run Supply Can Be Horizontal

A competitive industry’s long-run supply describes how much the industry supplies after firms have had time to enter or exit and adjust their operations. In a constant-cost industry, an increase in industry output does not raise firms’ input prices or otherwise change their production costs. As a result, demand can determine a different industry quantity without requiring a different long-run market price.

The key is to distinguish a firm’s output from the industry’s output. With free entry and exit, a change in demand can change the number of firms in the industry. In the standard constant-cost model, each active firm produces at its efficient long-run scale, and the number of firms adjusts to supply the quantity buyers demand at the long-run equilibrium price.

Definition: A constant-cost industry is a competitive industry in which expansion or contraction of industry output does not change the prices of the inputs firms use or otherwise shift their cost conditions. Under the standard assumptions of identical firms, free entry and exit, and competitive markets, the industry’s long-run supply curve is horizontal at the price equal to each firm’s minimum long-run average total cost.

This result depends on assumptions. Firms are price takers, use the same technology, and can enter or exit without barriers. Input prices remain constant as industry output changes. Firms choose output to maximize profit, and their long-run costs include the opportunity costs of the resources they use. The horizontal supply result applies to the long-run competitive equilibrium, not to the firm’s supply curve or to the industry’s short-run supply curve.

Economic profit is revenue minus all economic costs, including opportunity costs. Free entry and exit tend to eliminate positive and negative economic profit in long-run equilibrium. Zero economic profit does not mean that owners receive no return: it means their resources earn the normal return included in economic cost.

Deriving the Long-Run Equilibrium Price

For a firm producing positive output $q$, let $C(q)$ be its total cost, $MC(q)$ its marginal cost, and $ATC(q)=C(q)/q$ its average total cost. A profit-maximizing competitive firm chooses output where price equals marginal cost, when it is operating. In long-run equilibrium, entry and exit also require the firm to earn zero economic profit. Together, these conditions determine the firm’s long-run scale and the industry’s long-run price.

P=MC(q), \qquad P=ATC(q)

The first condition says that the last unit produced adds as much to revenue as to cost. The second says that revenue per unit equals total cost per unit, so economic profit is zero. Therefore, at the firm’s long-run equilibrium output, marginal cost equals average total cost. For a differentiable cost function, the reason is visible by differentiating average total cost:

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

At an interior minimum of average total cost, its derivative is zero. Since $q>0$, the expression above is zero exactly when $MC(q)=ATC(q)$. Thus the long-run price is the minimum of average total cost. This is the efficient scale: the output at which average total cost is as low as possible.

P^{LR}=\min_{q>0} ATC(q)

With unchanged costs, every firm has the same efficient scale and faces the same long-run equilibrium price. If demand increases, that price still covers the same minimum average cost. Entry adds firms, each operating at the efficient scale. If demand decreases, firms exit instead. Consequently, the market quantity can change while the long-run price remains at $P^{LR}$.

Theorem: Under the constant-cost assumptions, the long-run industry supply curve is horizontal at $P^{LR}$, the minimum average total cost of a representative firm. For any supported industry quantity, firms enter or exit as needed to supply that quantity at this price. This result does not say that the industry’s short-run supply curve is horizontal.

Worked Examples

Worked Example: Find the Long-Run Supply Price and Industry Quantities

Suppose identical competitive firms have the cost function $C(q)=q^2+4q+36$ for positive output. Input prices and technology do not change with industry size, and firms may enter or exit freely. Market demand is first $Q_D=280-10P$, and later it becomes $Q_D=340-10P$. Find the long-run price, each firm’s output, and the number of firms under each demand condition.

Method. First find the output that minimizes average total cost. Then use the unchanged long-run price to find market quantity from demand and divide it by each firm’s output. This avoids treating the market demand curve as an individual firm’s demand curve.

1. Find average total cost. For positive $q$,

ATC(q)=\frac{q^2+4q+36}{q}=q+4+\frac{36}{q}.

Differentiate and set the derivative equal to zero:

\frac{d\,ATC(q)}{dq}=1-\frac{36}{q^2}=0 \quad\Longrightarrow\quad q^2=36 \quad\Longrightarrow\quad q=6,

where only the positive output is relevant. At $q=6$,

ATC(6)=6+4+\frac{36}{6}=6+4+6=16.

Marginal cost is $MC(q)=2q+4$, so $MC(6)=12+4=16$. The marginal-cost and average-cost values agree at the minimum. Thus the long-run price is $P^{LR}=16$, and each active firm produces 6 units.

2. Calculate the first market quantity and firm count. At $P=16$, demand is

$$ Q_D=280-10(16)=280-160=120. $$

The implied number of firms is $N=120/6=20$. Check: $20$ firms producing $6$ units each supply $20(6)=120$ units.

3. Calculate the second market quantity and firm count. At the same long-run price,

Q_D=340-10(16)=340-160=180, \qquad N=\frac{180}{6}=30.

Check: $30(6)=180$. The change in demand raises long-run industry output from 120 to 180 and the number of firms from 20 to 30. The price remains 16 because the firms’ cost conditions are unchanged.

Worked Example: Diagnose Entry and Exit Incentives at Different Prices

Use the cost function $C(q)=q^2+4q+36$ from the preceding example. A firm faces a market price of either 20 or 14. For each price, find the firm’s profit-maximizing output and economic profit, then state the long-run entry or exit pressure.

Method. At each price, set price equal to marginal cost, $MC(q)=2q+4$, and compute revenue and total cost at that output. The mistake to avoid is inferring long-run equilibrium just because a firm is producing at the marginal-cost rule: the firm’s economic profit must also be checked.

At $P=20$. Solve $20=2q+4$, giving $2q=16$ and $q=8$. Revenue is $20(8)=160$. Total cost is

$$ C(8)=8^2+4(8)+36=64+32+36=132. $$

Economic profit is $160-132=28$. This is positive economic profit, so it creates an incentive for entry. The result is consistent with the long-run price of 16: price 20 is above the minimum average total cost.

At $P=14$. Solve $14=2q+4$, giving $2q=10$ and $q=5$. Revenue is $14(5)=70$. Total cost is

$$ C(5)=5^2+4(5)+36=25+20+36=81. $$

Economic profit is $70-81=-11$. Persistent losses create an incentive for long-run exit. At the long-run equilibrium price of 16, by contrast, each firm produces $q=6$, earns revenue $16(6)=96$, and incurs cost $C(6)=36+24+36=96$, for zero economic profit. Each calculation is consistent with the minimum average total cost of 16.

Worked Example: Recognize When Industry Costs Are Not Constant

Suppose firms have costs $C(q;w)=q^2+wq+36$, where $w$ is the price of a production input. Initially, $w=4$. If industry expansion raises the input price to $w=7$, compare the firms’ minimum average total costs before and after the increase. Explain what this implies for the long-run industry supply curve.

Method. Treat the input price as part of the firm’s cost conditions, and minimize average total cost separately at each value of $w$. The common trap is to assume a horizontal long-run supply curve merely because firms are competitive; that conclusion also requires costs to remain unchanged as the industry expands.

Average total cost is

ATC(q;w)=q+w+\frac{36}{q}.

Its derivative with respect to output is $1-36/q^2$. Setting it equal to zero gives $q^2=36$, so the efficient output is $q=6$ at either input price.

Initially, when $w=4$:

ATC(6;4)=6+4+\frac{36}{6}=6+4+6=16.

After the input price rises to $w=7$:

ATC(6;7)=6+7+\frac{36}{6}=6+7+6=19.

The minimum average total cost rises from 16 to 19. If industry expansion raises the input price, costs are not constant across industry sizes, so the horizontal-supply conclusion does not apply over that expansion. The result illustrates why unchanged input costs are a substantive assumption, not a consequence of perfect competition alone.

Reading the Long-Run Supply Result Correctly

The horizontal curve is a relationship between the industry’s long-run price and total quantity supplied. It is not a claim that each firm can supply any quantity at the same price. At the long-run equilibrium price, each firm produces its efficient-scale output; industry quantity changes through the number of active firms.

A useful way to organize the result is to separate the firm-level and industry-level decisions:

Find the efficient firm scale.
Minimize average total cost to obtain the output at which the firm’s long-run unit cost is lowest.

Set the long-run price.
In competitive long-run equilibrium, price equals minimum average total cost, so firms earn zero economic profit.

Find industry quantity.
At that price, use market demand to determine the total quantity buyers demand.

Determine the number of firms.
Divide industry quantity by the efficient output per firm. Entry or exit adjusts the number of active firms.

The model commonly treats the number of firms as adjustable enough to supply the required industry quantity. If firms must be indivisible and the number of firms is restricted to whole numbers, exact quantities may not be attainable at every point. The horizontal curve is the standard competitive model’s long-run result, often understood as applying to a large industry where such indivisibilities are negligible.

The result matters because it shows how industry size can change without permanently changing the price when expansion does not affect costs. This can help organize analysis of competitive markets, including hypothetical markets for products such as insurance, provided the constant-cost and entry assumptions are appropriate. In applying the model, check which level of analysis a question asks about: firm output, industry quantity, or the number of firms. The long-run price alone does not specify all three.

Common Mistakes

Using the wrong cost benchmark. The long-run price is minimum average total cost, not minimum marginal cost. Use the point where average total cost is minimized.
Confusing zero economic profit with zero accounting profit. Zero economic profit includes the normal return to the firm’s resources; it does not mean those resources earn nothing.
Changing firm output when only industry demand has changed. Under unchanged costs, the efficient output per firm stays the same. Entry or exit changes the number of firms.
Calling every competitive industry’s long-run supply curve horizontal. The result requires input prices and firms’ cost conditions to remain unchanged as industry output changes.
Applying the long-run result to the short run. A horizontal long-run industry supply curve does not establish that the short-run industry supply curve is horizontal.

Key takeaway: In a constant-cost competitive industry with free entry and exit, the long-run price equals firms’ minimum average total cost. At that price, each firm produces at its efficient scale, while entry or exit adjusts the number of firms to match industry demand.

Check Your Understanding

For each problem, use the long-run constant-cost framework only when its assumptions are stated.

Identical firms have $C(q)=q^2+9$, and the industry is constant-cost with free entry and exit. Find the efficient output per firm and the long-run equilibrium price. (Answer: $q=3$; $P^{LR}=6$.)
A firm has $C(q)=2q^2+8q+50$ at every industry size. Find the output that minimizes average total cost and the corresponding long-run price. Verify that marginal cost equals average total cost there. (Answer: $q=5$; $P^{LR}=28$; $MC(5)=ATC(5)=28$.)
Identical firms have $C(q)=q^2+4q+36$. The industry is constant-cost, and demand is $Q_D=500-20P$. Find the long-run market price, industry quantity, and number of firms. (Answer: $P^{LR}=16$; $Q=180$; $N=30$.)
For a firm with $C(q)=q^2+4q+36$, market price is 18. Find its profit-maximizing output and economic profit. Does this price create an entry or exit incentive? (Answer: $q=7$; economic profit $=\$13$; entry incentive.)
For firms with $C(q;w)=q^2+wq+36$, expansion raises the input price from $w=4$ to $w=7$. Find minimum average total cost at each input price and state whether the constant-cost horizontal-supply conclusion applies to that expansion. (Answer: minimum average total cost rises from 16 to 19; no, the expansion changes firms’ costs.)

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