Natural Monopoly and Economies of Scale

Why the Number of Suppliers Can Depend on Cost

Some industries require a large network, facility, or other costly capacity before serving customers. If that resource can serve many customers at once, one supplier may be able to provide the market’s total output at lower cost than several suppliers duplicating the same infrastructure. This cost pattern is called a natural monopoly. The key evidence is not simply that a firm is large or that competitors are absent: it is the relationship between production cost and the amount of output served.

This tutorial is aligned with the SOA VEE Economics (Validation by Educational Experience) syllabus. It focuses on long-run cost comparisons. A natural monopoly is a statement about the cost of serving a market, not a conclusion about what price a supplier charges or how much it produces.

Definition: A natural monopoly exists over a specified market or range of output when one supplier can produce the market’s total output at a cost no greater than the cost of any division of that output among multiple suppliers. In symbols, for total output $Q$ and any division $Q=q_1+\cdots+q_n$, the cost condition is $C(Q)\leq \sum_{j=1}^{n}C(q_j)$. The cost function must describe the relevant technology and time horizon.

The definition is about total cost, not simply the number of firms or the size of a single firm’s fixed cost. A comparison between one firm and one particular two-firm division can illustrate the idea, but by itself does not establish the condition for every possible division. When average cost is falling throughout the relevant output range, however, the cost comparison follows directly, as shown below.

Economies of Scale, Average Cost, and Marginal Cost

For a firm producing $q>0$ units, average cost is total cost per unit, $AC(q)=C(q)/q$. Marginal cost is the change in total cost from producing a small additional quantity; for a differentiable cost function, $MC(q)=C'(q)$. In long-run analysis, the cost function allows the firm to adjust all inputs, including its scale of operation.

Definition: A technology has economies of scale over an output range when long-run average cost falls as output increases over that range. A natural monopoly has a cost advantage from serving the market as one supplier when the relevant cost function makes the total cost of one supplier no greater than the total cost of any alternative division of market output.

When cost is differentiable, the relationship between marginal and average cost explains whether average cost is rising or falling. Differentiating $AC(q)=C(q)/q$ gives

AC'(q)=\frac{qC'(q)-C(q)}{q^2} =\frac{MC(q)-AC(q)}{q}.

Since $q>0$, the sign of $AC'(q)$ is the sign of $MC(q)-AC(q)$. Thus, if $MC(q)<AC(q)$, average cost is falling at that output; if $MC(q)>AC(q)$, it is rising. Equality means the derivative of average cost is zero at that point, though it does not by itself establish whether average cost has a minimum there.

Why falling average cost implies the cost comparison: Suppose $AC(q)$ is falling for all positive quantities up to market output $Q$. In any division $Q=q_1+\cdots+q_n$, each $q_j\leq Q$, so $AC(q_j)\geq AC(Q)$. Therefore,

\sum_{j=1}^{n} C(q_j) =\sum_{j=1}^{n} q_j AC(q_j) \geq \sum_{j=1}^{n} q_j AC(Q) =QAC(Q) =C(Q).

This proves that one supplier’s cost is no greater than the cost of any such division when average cost falls throughout the relevant range. The conclusion depends on that range: falling average cost up to $Q$ supports the result for a market of size $Q$, but does not establish it for larger markets if average cost stops falling beyond $Q$.

Comparing One Supplier With Several

For an explicit comparison, compute the cost of producing $Q$ as one firm and then compute total cost under the proposed division. Do not multiply a per-unit cost by total market output without checking which quantity that per-unit cost describes. If firms have identical cost functions, each firm’s cost is evaluated at its own output.

Worked Example: A Shared Network With a Fixed Cost

A hypothetical network service has long-run cost $C(q)=400+20q$ dollars for any positive output $q$. The relevant market requires $Q=40$ service units. Compare one supplier with two identical suppliers dividing output equally. Find average and marginal cost at the one-supplier output, and state what the comparison establishes.

Method. Evaluate the cost function at each supplier’s output, then add supplier costs. This avoids confusing one firm’s average cost with the cost of the entire market.

One supplier. At $q=40$, $C(40)=400+20(40)=400+800=\$1{,}200$. Its average cost is $AC(40)=\$1{,}200/40=\$30$ per unit. Since $C'(q)=20$, marginal cost at $q=40$ is $\$20$ per additional unit.

Two equal suppliers. Each produces $40/2=20$ units. Each incurs $C(20)=400+20(20)=400+400=\$800$. Together they incur $2(\$800)=\$1{,}600$. The one-supplier cost is lower by $\$1{,}600-\$1{,}200=\$400$.

Check and interpretation. At output 40, marginal cost is below average cost, consistent with falling average cost. In fact, $AC(q)=400/q+20$ falls for every $q>0$; the proof above therefore shows that one supplier costs no more than any division of these 40 units, not just the equal two-firm division. The common mistake is to compute $400+20(40)$ as $\$800$; the correct sum is $\$1{,}200$.

A fixed cost can help explain why duplicating infrastructure is costly, but the phrase “high fixed cost” alone is not a complete test for natural monopoly. Variable costs and the market’s required output also matter. The relevant question is whether one supplier’s total cost compares favorably with the costs of supplying the same total output using multiple suppliers.

Worked Example: Checking the Relevant Output Range

A hypothetical service has cost $C(q)=600+10q+0.1q^2$ dollars for positive output. Market demand requires $Q=60$ units. Determine whether average cost falls over the range through 60, calculate the single-supplier cost, and compare it with two equal suppliers.

Method. Use the average-cost derivative to check economies of scale over the whole relevant range, then calculate the two total costs. This distinguishes a range-specific conclusion from an unsupported claim about every possible market size.

Average cost is $AC(q)=600/q+10+0.1q$. Its derivative is $AC'(q)=-600/q^2+0.1$. For $0<q\leq60$, $q^2\leq3{,}600$, so $600/q^2\geq600/3{,}600=1/6$. Hence $AC'(q)\leq-1/6+0.1=-1/15<0$: average cost falls throughout the relevant range.

One supplier. $C(60)=600+10(60)+0.1(60^2)=600+600+360=\$1{,}560$. Thus $AC(60)=\$1{,}560/60=\$26$ per unit. Marginal cost is $C'(q)=10+0.2q$, so $MC(60)=10+12=\$22$ per unit, below average cost.

Two equal suppliers. Each produces 30 units. Each incurs $C(30)=600+10(30)+0.1(30^2)=600+300+90=\$990$. Combined cost is $2(\$990)=\$1{,}980$, which is $\$1{,}980-\$1{,}560=\$420$ more than one supplier’s cost.

Check and interpretation. The calculation and the falling-average-cost proof together establish that one supplier is no more costly than any division of the 60 units. They do not establish the same conclusion above 60 units: $AC'(q)$ becomes zero at $q=\sqrt{6{,}000}$, approximately 77.46, and turns positive for larger output. The trap is to infer that a natural monopoly must have falling average cost at every conceivable scale; the conclusion is tied to the relevant market range.

Natural Monopoly Is a Cost Condition, Not an Entry Barrier

Entry barriers and natural monopoly are related to market structure, but they describe different things. A natural monopoly is defined by the cost advantage of one supplier over multiple suppliers. Other entry barriers restrict or discourage new firms for reasons such as a legal exclusive right, control of an essential input, or strategic conduct. Those conditions can exist without falling average cost or cost subadditivity.

Conversely, a cost structure can favor one supplier even if no law prohibits entry. A potential entrant might still be able to enter, but dividing the market between firms could raise total production cost because each would have to duplicate a costly network. A natural-monopoly cost advantage therefore does not, by itself, prove that entry is legally blocked, that an incumbent behaves competitively, or that its price is low.

Question	Natural-monopoly cost condition	Other entry barrier
What is being assessed?	Whether one supplier can serve the market at no greater total cost than multiple suppliers	Whether a condition makes entry difficult or restricted
Evidence to examine	Cost function, output range, and possible divisions of market output	For example, legal restrictions or control of an essential input
What does it establish?	A cost advantage to serving the specified market as one supplier	A constraint on actual or potential entry

Worked Example: Separating Cost Advantage From a Legal Restriction

A hypothetical local distribution service has cost $C(q)=900+6q$ dollars for any positive output. The market requires 100 units. Calculate costs for one supplier and two equal suppliers. Then consider a separate rule granting one operator an exclusive license: explain what the cost calculations show and what the license adds.

Method. Calculate the production-cost comparison first, then assess the legal condition separately. This avoids treating a legal restriction as evidence about the cost curve.

One supplier. $C(100)=900+6(100)=900+600=\$1{,}500$. Average cost is $\$1{,}500/100=\$15$ per unit, and marginal cost is $C'(q)=\$6$ per unit.

Two equal suppliers. Each produces 50 units, so each incurs $C(50)=900+6(50)=\$1{,}200$. Combined cost is $2(\$1{,}200)=\$2{,}400$, which exceeds the single-supplier cost by $\$900$. Average cost, $AC(q)=900/q+6$, falls at every positive output. Thus, for this cost function and this market size, the falling-average-cost proof establishes the cost advantage of one supplier over any division of the 100 units.

Interpretation and check. The cost function shows a natural-monopoly cost condition over this market. The exclusive license is a separate legal entry barrier; it does not cause the calculated cost advantage. Neither calculation establishes what price is charged. The common mistake is to use the license itself as proof of economies of scale, when that conclusion must come from cost information.

What to Establish in a Cost Comparison

A reliable analysis makes its scope explicit. First identify the market output and the cost function that applies to each supplier. Then determine whether average cost falls throughout the range needed for the comparison, or directly compare costs for the divisions specified. If average cost is not falling across the relevant range, do not assume that one supplier is least costly; examine the actual cost sums instead.

Keep three conclusions separate: a declining average-cost curve indicates economies of scale over a range; cost subadditivity says one supplier costs no more than every division of a specified total output; and an entry barrier is a condition that restricts or discourages entry. Falling average cost over the full interval from positive output to market quantity is sufficient for cost subadditivity there, but evidence about only one split is not sufficient to establish the universal comparison.

Use the same total market output in every comparison.
For multiple suppliers, evaluate each firm’s cost at its own output and add the results.
Use $MC<AC$ as evidence that average cost is falling at that point, not as proof by itself that it falls over the whole market range.
State the range and technology to which the natural-monopoly conclusion applies.
Do not infer a price, legal prohibition on entry, or a particular market outcome from the cost condition alone.

Key takeaway: A natural monopoly is identified by cost comparisons for a specified market. Falling average cost over the relevant range is a useful sufficient condition: each smaller supplier then has average cost at least as high as the single supplier serving the full market, so dividing production cannot lower total cost.

Check Your Understanding

For each hypothetical cost function, use the stated market output and show the relevant cost comparison before checking the final answer.

A service has $C(q)=300+8q$ dollars and market output $Q=30$. Find one supplier’s total and average cost, then compare total cost with two equal suppliers. (Answer: One supplier costs $\$540$, with average cost $\$18$; two equal suppliers cost $\$840$, so one supplier costs $\$300$ less.)
A firm’s cost is $C(q)=500+4q$ dollars and the market requires $Q=80$. Calculate average and marginal cost at 80, and compare one supplier with two equal suppliers. (Answer: One supplier costs $\$820$, $AC(80)=\$10.25$, and $MC(80)=\$4$; two equal suppliers cost $\$1{,}320$, so one costs $\$500$ less.)
A provider has $C(q)=200+2q+0.05q^2$ dollars, and market output is $Q=40$. Find one supplier’s cost and marginal cost, compare with two equal suppliers, and determine whether average cost is falling at 40. (Answer: One supplier costs $\$360$, with $MC(40)=\$6$ and $AC(40)=\$9$; two equal suppliers cost $\$520$; average cost is falling at 40.)
A network operator has $C(q)=1{,}000+5q$ dollars, with market output $Q=100$. Compare one supplier with four equal suppliers and explain whether average cost declines with output. (Answer: One supplier costs $\$1{,}500$; four suppliers producing 25 each cost $\$4{,}500$ total; $AC(q)=1{,}000/q+5$ declines for positive $q$, so one supplier has the cost advantage over this market.)
A service has $C(q)=400+10q$ dollars and market output $Q=20$. A law also grants one provider an exclusive license. Calculate costs for one and two equal suppliers, and distinguish the two findings. (Answer: One supplier costs $\$600$; two equal suppliers cost $\$1{,}000$, so one has a $\$400$ cost advantage. The declining average cost supports a natural-monopoly cost condition at this output; the exclusive license is a separate legal entry barrier.)

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