Natural monopoly

William Baumol (1977)[2] 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. 810). Baumol linked the definition to the mathematical concept of subadditivity; specifically, subadditivity of the 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.[3]

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

Proof

Consider n output vectors ${\displaystyle v_{1}y,...,v_{n}y}$, when ray average costs are strictly declining: ${\displaystyle C(\sum v_{j}y)/\sum v_{j}<C(\sum v_{i}y)/\sum v_{i}}$ Therefore: ${\displaystyle {\frac {C(\sum v_{j}y)}{\sum v_{j}}}<\sum {\frac {v_{i}}{\sum v_{j}}}{\frac {C(v_{i}y)}{v_{i}}}={\frac {\sum C(v_{i}y)}{\sum v_{j}}}}$ Which gives: ${\displaystyle C(\sum v_{j}y)<\sum C(v_{i}y)}$ Therefore the cost function is strictly subadditivite.

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

Proof

Let ${\displaystyle c_{1}<c_{2}}$ in the piecewise-linear cost function: ${\displaystyle C(y)={\begin{cases}c_{1},&y<y_{B}\\c_{2},&y>y_{B}\end{cases}}}$ Let ${\displaystyle y_{r}<y_{b}<y_{s}}$. The cost function is not concave, average cost increases after ${\displaystyle y_{b}}$ and ray average cost is greater at ${\displaystyle y_{s}}$ than ${\displaystyle y_{r}}$. Also: total cost of any output y by a single firm ${\displaystyle \leq c_{2}<2c_{1}\leq }$ 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.

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.

Proof
Let ${\displaystyle 0<a<1}$ and ${\displaystyle 0<k<1/2}$ in the cost function for two outputs: ${\displaystyle C=y_{1}^{a}+y_{1}^{k}y_{2}^{k}+y_{2}^{a}}$ C is strictly concave and not subbaditive: ${\displaystyle C(1,1)=3>C(1,0)+(C,0,1)=2}$

Therefore:

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

A cost function c is subadditive at an output x if ${\displaystyle {\displaystyle {\begin{aligned}c(x)&=c(x^{1})+c(x^{2})+...+c(x^{k})\end{aligned}}}}$ such that ${\displaystyle \sum _{i=1}^{k}x^{i}=x}$, 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.

1. Railways:
   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.
2. Telecommunications and Utilities:
   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.

A wave of nationalisation across Europe after World War II created state-owned companies in each of these areas, many of which operate internationally bidding on utility contracts in other countries. However, this approach can raise its own problems. In the past, some governments have used the state-provided utility services as a source of cash flow for funding other government activities, or as a means of obtaining hard currency. As a result, governments seeking funding began to seek other solutions, namely regulation and providing services on a commercial basis, often through private participation.[15]

In recent years, bodies of information have observed the correlation between utility subsidies and welfare improvements.[16] Today, across the world, public utilities are widely used to provide state-run water, electricity, gas, telecommunications, mass-transportation and postal services.

Alternatives to a state-owned response to natural monopolies include both open source licensed technology and co-operatives management where a monopoly's users or workers own the monopoly. For instance, the web's open-source architecture has both stimulated massive growth and avoided a single company controlling the entire market. The Depository Trust and Clearing Corporation is an American co-op that provides the majority of clearing and financial settlement across the securities industry ensuring they cannot abuse their market position to raise costs. In recent years a combined cooperative and open-source alternative to emergent web monopolies has been proposed, a platform cooperative,[17] where, for instance, Uber could be a driver-owned cooperative developing and sharing open-source software.[18]