The properties of a natural monopoly are as follows.
Fixed costs are very large relative to their variable costs. Therefore, average costs are very large at small amounts of output and fall as output increases. Thus, average costs exceed marginal costs over a wide range of output.
Average costs exceed marginal costs over the "relevant range of output" (i.e., the range between the first unit of output and the amount consumers would demand at a zero price). Therefore, average costs continue to fall over the relevant range of output.
As a result, one firm, a natural monopoly, can provide a given amount of output at a lower average cost than could several competing firms.
There are at least four policies the government could follow in regards to a natural monopoly.
Allow the monopoly to maximize profits by producing at the monopoly level. This results in a deadweight loss.
Require the monopoly to set its price where the average cost curve crosses the demand curve. This transfers some surplus from the monopoly to consumers, expands output, increases social surplus, and reduces deadweight loss.
Require the monopoly to set its price where the marginal cost curve crosses the demand curve. This eliminates deadweight loss but revenues no longer cover costs. As a result, tax money must be used to subsidize the production of the good.
Require the monopoly to charge a zero price. This also results in a deadweight loss and causes costs to exceed revenues, necessitating subsides.