A pure monopoly is a firm that satisfies the following conditions:
In practice, pure monopolies are very rare. For instance, a supermarket may be the only food supplier in a particular town, but if it raises its prices and retains too much of a profit, a competitor may enter the space. Even the threat of serious competition entering the market forces the existing firm to act conscionably, and differently from how it would act otherwise. A train company may be the only carrier in a particular station, but if cars are also available in the area, there exists a close substitute to the output good.
A natural monopoly is a firm with such extreme economies of scale that once it begins creating a certain level of output, it can produce more at a far lower cost than any smaller competitor. Natural monopolies exist far more frequently than pure monopolies, mainly because the requirements are not as stringent.
Natural monopolies occur when, for whatever reason, the average cost curves decline over a relevant span of output quantities. A firm with high fixed costs relative to its marginal costs will have declining average costs for a significant span of quantities. A firm with a decreasing marginal cost structure will also have declining average costs. For example, utilities and software are two industries where natural monopolies occur often.
A monopoly differs from competitive firms in that it is not a price taker. Because it is the only supplier in the market, it faces a downward sloping demand curve, the market demand curve. As a result, the monopoly is free to choose its price and quantity according to market demand.
Monopolies are still profit maximizing firms and are thus going to satisfy the profit maximizing condition that marginal cost equal marginal revenue. The key to understanding monopolies and monopoly power is the marginal revenue calculation. In a perfectly competitive market, there exists a market price. Marginal revenue is simply equal to price in this market; every additional unit that is sold brings the market price. In a monopoly, however, every quantity is associated with a different price. The marginal revenue is not simply the price.