Natural monopoly doesn't have anything to do with nature. 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. Economies of scale occur when, as a firm produces more of a product, their average costs of production fall. Economies of scale aren't uncommon (here's a bunch of examples), but for natural monopoly to arise, the economies have to be large. This happens 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. When this is the case, the average cost (AC) and marginal cost (MC) curves look something like this:
Examples of industries with this type of cost structure include anything with a large up-front infrastructure cost (water supply, electricity generation and supply, telecommunications, rail, roads, mail, etc.), but potentially lots of IT-based industries as well (search engines, instant messaging, social networks, etc.). This Ars Technica article from last month provides a good example from the U.S., in the form of Internet Service Providers (ISPs). According to the article:
A new fiber provider needs a slew of government permits and construction crews to bring fiber to homes and businesses. It needs to buy Internet capacity from transit providers to connect customers to the rest of the Internet. It probably needs investors who are willing to wait years for a profit because the up-front capital costs are huge. If the new entrant can't take a sizable chunk of customers away from the area's incumbent Internet provider, it may never recover the initial costs. And if the newcomer is a real threat to the incumbent, it might need an army of lawyers to fend off frivolous lawsuits designed to put it out of business.So, here is the problem. There is a large up-front cost associated with setting up an ISP (or expanding your ISP into a new location), because the would-be ISP needs government permits (expensive) and needs to lay the cables necessary to connect their customers (very expensive - the article estimates that Google spent US$84 million to build a fibre network that went past 149,000 homes in Kansas City, without including the costs of actually connecting any of the homes, an average cost of around $560 per home if all of them connected). All of these costs need to be paid before even one customer can be connected to the ISP. Add onto that the cost of defending lawsuits, and that is a pretty big up-front (fixed) cost.
If the new entrant ISP has only a small subscriber base, this fixed cost is spread over only a small number of customers, meaning that their average costs will be very high. Compare that with an incumbent ISP which already has many subscribers, spreading their fixed cost over a large number of customers and leading to lower average costs. The incumbent ISP could easily keep prices at a point where the new entrant ISP would be making a loss, eventually running out of money and closing down.
Of course, you might note that predatory pricing is illegal (including in New Zealand). However, the incumbent ISP wouldn't need to engage in predatory pricing here. The price they set would not need to be artificially low (i.e. below cost), in order to force the new entrant out. The incumbent ISP could continue to make a profit (albeit a smaller profit than before) while the new entrant ISP makes losses.
So, we are probably left with a single (natural monopoly) firm serving the market. The main problems with monopoly providers are that they charge a high price (they are profit maximising, so they produce the quantity where MR=MC, i.e. QM on the diagram below, and sell it as the price PM), which limits quantity below the socially-efficient amount of the product (where AR=MC, i.e. at QS on the diagram below, with the price PS).
How can government solve the problem of natural monopoly then?
One potential solution is to regulate price. By introducing a price ceiling (a legal maximum price), you can force the monopoly to charge a lower price than they otherwise would (ultimately you could set the price ceiling at PS), which increases welfare. The first problem is that if you set the price ceiling at the price that maximises welfare (PS), then the natural monopoly will make a loss (since PS is less than the firm's average costs CS) and may choose to shut down (leaving you with no market for the good at all). An alternative is to set the price ceiling high enough that the firm makes no loss (price ceiling of at least CS) - this won't maximise welfare, but it will increase welfare above the monopoly pricing situation. However, even if the natural monopoly makes no profit or a small profit, there are still problems with regulating price. The main problem is that it is pretty inflexible and won't readily adjust to changes in the market. For instance, if demand increases substantially, we would expect price to increase but in this case it is held low by the price ceiling. This would lead to under-investment in new capacity by the monopoly, and decreases in service quality, etc.
A second solution is for the government to own the natural monopoly itself. Government ownership of natural monopolies is common in many countries (think about rail, telecommunications, water supply, electricity, etc.). That way, the government can charge whatever price it wants, and can ensure that economic welfare is increased (again, at the expense of profits). An advantage of government ownership is that the government can usually borrow more cheaply than private firms, which means that the costs of paying for the large infrastructure investment to set up the natural monopoly firm (in the case of utilities, for example) are lower. The government also doesn't need to worry about collateral - consider this bit from the Ars Technica article:
"One of the really terrible things about being in this business is the infrastructure you're building is very expensive, but it has no collateral value for the bank," Montgomery also said. "If I put $1 million of fiber in the ground and go to the bank, it'll say, 'ok I'm going to need a million dollars' worth of collateral. The fiber isn't worth anything to us, so you're going to have to cough up something else, gold bars, cash, something.'"However, government ownership is generally less efficient that private ownership. Private firms must answer to shareholders, who expect profits. So private firms have incentives to seek gains in efficiency. Government-owned firms have weaker incentives to seek gains in efficiency (this is a type of X-inefficiency). So, in the long run government ownership may actually make society worse off, and the best option may be for government to set up the natural monopoly (and take advantage of the low borrowing costs), then privatise.
A third solution is to find some way of ensuring your market has the benefits of competition (so firms will compete on price, service quality, etc.), without the necessity of firms incurring multiple instances of the fixed costs. One way of achieving this (in the case of utilities) is to share the infrastructure between many firms. Essentially, you separate the competitive parts of the business (usually retail) from the natural monopoly parts (usually the infrastructure itself). For instance, with ISPs you have one set of fibre linking to all houses in an area but the fibre is not owned by any of the competing firms. Instead, the fibre network may be owned by the government, or by a single firm regulated by the government, and access to the network is provided to all ISPs on the same cost basis. This gives none of the firms a cost advantage over any other, and all contribute to the costs of network maintenance, etc.
Of course, this last solution is not without its problems. The natural monopoly still exists (in the form of the firm that owns the network infrastructure itself), and will need to be regulated using one of the previous two options). However, the consumers are to some extent insulated from the activities of the natural monopoly and have competition in the marketplace for the product, and benefit from the resulting lower prices, better service quality, etc. This last solution is essentially what we have in place in the electricity market in New Zealand, and in the roll-out of ultra-fast broadband.