I picked up this natural monopoly theory from University of Waikato Senior Economics Lecturer Michael Cameron on his excellent blog entitled Sex, Drugs and Economics. This is very useful for A2 students when studying market structures.
A natural monopoly arises where one producer of a product is so much more efficient (by efficient I mean they produce at lower cost) than many suppliers that new entrants into the market would find it difficult, if not impossible, to compete with them. It is this cost advantage that creates a barrier to entry for other firms, and leads to a monopoly. Natural monopolies typically arise where there are large economies of scale (when, as a firm produces more of a product, their average costs of production fall). Economies of scale are common when there is a very large up-front (fixed) cost of production, and the marginal costs (the cost of supplying an additional unit of the product) are small (the cost structure is shown in the figure below, with a simplifying assumption that the marginal cost of production is low and constant). The markets for utilities, where the up-front cost includes the cost of having all of the infrastructure in place, are good examples. Rail is another example, since you need the tracks, the rolling stock, and the associated stations and other buildings in place before you can start to provide rail services.
Now natural monopolies, like other firms, are assumed to be profit maximisers. That is, they will operate at the point where marginal revenue is equal to marginal cost. That is, they will operate at the price PM and the quantity QM in the diagram above. At that point, the producer surplus is the area PMBHPS, while the firm’s profit is the area PMBKL (the difference between profit and producer surplus arises because of the large up-front fixed costs, which are subtracted from profits, but not from producer surplus). However, consumer surplus in this market is GBPM, and total welfare is GBHPS. This leaves a deadweight loss equal to the area BEH.
Now, if the government owned the natural monopoly, it doesn’t necessarily have to profit maximise if it doesn’t want to. Government could choose to maximise total welfare instead. It would do this by setting the price at the point where marginal social benefit is equal to marginal social cost. That is, the market will operate at the price PS and the quantity QS. At that point, producer surplus is zero (since every unit is sold for marginal cost), but the profit is negative (JDEPS) because price is below average cost. On the other hand, consumer surplus is GEPS, and total welfare is maximised at GEPS.