Editing Natural monopoly (section)

== Definition ==
Two different types of cost are important in [[microeconomics]]: [[marginal cost]] and [[fixed cost]]. The marginal cost is the cost to the company of serving one more customer. In an industry where a natural monopoly does not exist, the vast majority of industries, the marginal cost decreases with economies of scale, then increases as the company has growing pains (overworking its employees, bureaucracy, inefficiencies, etc.). Along with this, the average cost of its products decreases and increases. A natural monopoly has a very different cost structure. A natural monopoly has a high fixed cost for a product that does not depend on output, but its marginal cost of producing one more good is roughly constant, and small.

It is generally believed that there are two reasons for natural monopolies: one is economies of scale, and the other is economies of scope.

[[File: Natural monopoly.jpg|thumb|right|270px|A graphical explanation of the inefficiencies of having several competitors in a naturally monopolistic market. AC = average cost (per customer), D = demand.]]

All industries have costs associated with entering them. Often, a large portion of these costs is required for [[investment (macroeconomics)|investment]]. Larger industries, like utilities, require an enormous initial investment. This [[barrier to entry]] reduces the number of possible entrants into the industry regardless of the earning of the corporations within. The production cost of an enterprise is not fixed, except for the effect of technology and other factors; even under the same conditions, the unit production cost of an enterprise can also tend to decrease with the increase in the total production output. The reason is that the actual product of the enterprise as it continues to expand, the original fixed costs are gradually diluted. This is particularly evident in companies with significant fixed-cost investments. Natural monopolies arise where the largest supplier in an industry, often the first supplier in a market, has an overwhelming cost advantage over other actual or potential competitors; this tends to be the case in industries where fixed costs predominate, creating economies of scale that are large in relation to the size of the market, as is the case in water and electricity services. The fixed cost of constructing a competing transmission network is so high, and the marginal cost of transmission for the incumbent so low, that it effectively bars potential competitors from the monopolist's market, acting as a nearly insurmountable barrier to entry into the market place.

A firm with high fixed costs requires a large number of customers in order to have a meaningful return on investment. This is where economies of scale become important. Since each firm has large initial costs, as the firm gains market share and increases its output the fixed cost (what they initially invested) is divided among a larger number of customers. Therefore, in industries with large initial investment requirements, [[average total cost]] declines as output increases over a much larger range of output levels.

In real life, companies produce or provide single goods and services but often diversify their operations. Suppose the cost of having multiple products by one enterprise is lower than making them separately by several enterprises. In that case, it indicates that there is an economy of scope. Since the unit product price of a company that produces a specific product alone is higher than the corresponding unit product price of a joint production company, the companies that make it separately will lose money. These companies will either withdraw from the production field or be merged, forming a monopoly. Therefore, well-known American economists Samuelson and Nordhaus pointed out that economies of scope can also produce natural monopolies.

Companies that take advantage of economies of scale often run into problems of bureaucracy; these factors interact to produce an "ideal" size for a company, at which the company's average cost of production is minimized. If that ideal size is large enough to supply the whole market, then that market is a natural monopoly.

Once a natural monopoly has been established because of the large initial cost and that, according to the rule of economies of scale, the larger corporation (to a point) has a lower average cost and therefore an advantage over its competitors. With this knowledge, no firms will attempt to enter the industry and an [[oligopoly]] or monopoly develops.

=== Formal definition ===
[[William Baumol]] (1977)<ref>Baumol, William J., 1977. "On the Proper Cost Tests for Natural Monopoly in a Multiproduct Industry", ''American Economic Review'' 67, 809–22.</ref> provides the current formal definition of a natural monopoly. He defines a natural monopoly as "[a]n industry in which multi-firm production is more costly than production by a monopoly" (p.&nbsp;810). Baumol linked the definition to the mathematical concept of [[subadditivity]]; specifically, subadditivity of the [[cost curve|cost function]]. Baumol also noted that for a firm producing a single product, scale economies were a sufficient condition but not a necessary condition to prove subadditivity, the argument can be illustrated as follows:

Proposition: Strict economies of scale are sufficient but not necessary for ray average cost to be strictly declining.<ref>W. J. Baumol, 1976. "Scale Economies, Average Cost and the Profitability of Marginal-Cost Pricing"</ref>

Proposition: Strictly declining ray average cost implies strict ray subadditivity.

:{| class="toccolours collapsible collapsed" width="90%" style="text-align:left"
!Proof
|-
|Consider n output vectors <math>v_1y,...,v_ny</math>, when ray average costs are strictly declining:

<math>C(\sum v_jy)/\sum v_j < C(\sum v_iy)/\sum v_i</math>

Therefore:

<math>\frac{C(\sum v_jy)}{\sum v_j}<\sum \frac {v_i} {\sum v_j} \frac{C(v_iy)}{v_i}=\frac{\sum C(v_iy)}{\sum v_j}</math>

Which gives:

<math>C(\sum v_jy)<\sum C(v_iy)</math>

Therefore the cost function is strictly subadditivite.
|}

Proposition: Neither ray concavity nor ray average costs that decline everywhere are necessary for strict subadditivity.

:{| class="toccolours collapsible collapsed" width="90%" style="text-align:left"
!Proof
|-
|Let <math>c_1 < c_2</math> in the piecewise-linear cost function:

<math>C(y) = \begin{cases} c_1, & y < y_B \\ c_2, & y > y_B \end{cases}</math>

Let <math>y_r < y_b < y_s</math>. The cost function is not concave, average cost increases after <math>y_b</math> and ray average cost is greater at <math>y_s</math> than <math>y_r</math>. Also:

total cost of any output y by a single firm <math> \le c_2 < 2 c_1 \le</math> total cost of production by more than one firm

Therefore the cost function is strictly subadditivite.
|}

Combining all propositions gives:

Proposition: Global scale economies are sufficient but not necessary for (strict) ray subadditivity, the condition for natural monopoly in the production of a single product or in any bundle of outputs produced in fixed proportions.

==== Multiproduct case====

On the other hand if firms produce many products scale economies are neither sufficient nor necessary for subadditivity:

Proposition: Strict concavity of a cost function is not sufficient to guarantee subadditivity.

:{| class="toccolours collapsible collapsed" width="90%" style="text-align:left"
!Proof
|-
|Let <math>0 <a <1</math> and <math>0 < k <1/2 </math> in the cost function for two outputs:

<math>C=y_1^a+y_1^k y_2^k+y_2^a</math>

C is strictly concave and not subbaditive:
<math>C(1,1)=3>C(1,0)+C(0,1)=2</math>
|}

Therefore:

Proposition: Scale economies are neither necessary nor sufficient for subadditivity.

=== Mathematical notation of subadditivity ===
A cost function ''c'' is ''subadditive'' at an output ''x'' if <math>{\displaystyle {\begin{aligned}c(x)&= c(x^1)+c(x^2)+...+c(x^k)\end{aligned}}} </math> such that  <math>
\sum^{k}_{i=1} x^i = x</math>, with all x being non-negative. In other words, if all companies have the same production cost function, the one with the better technology should monopolize the entire market such that the total cost is minimized, thus causing natural monopoly due to its technological advantage or condition.

=== Examples ===
#[[Railways]]:<br/>The costs of laying tracks and building networks coupled with that of buying or leasing the trains prohibits or deters the entry of any competitor. Rail transport also fits other characteristics of a natural monopoly because it is assumed to be an industry with significant long run economies of scale.
#[[Telecommunications]] and [[Utilities]]:<br/>The costs of building telecommunication poles and growing a cell network would just be too exhausting for other competitors to exist. Electricity requires grids and cables whilst water services and gas both require pipelines whose costs are just too high to be able to have existing competitors in the public market. However, natural monopolies are usually regulated and they face increasing competition from private networks and specialty carriers.