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It's also difficult to consider without defining a lot of the terms you are talking about. In this case, "pure monopoly" could take a few meanings, and you have the same issue with "compete." The problem with these definitions stems from a couple of ideas:
1) Monopolies have often been temporary
2) Monopolies generally only take control of a certain region
Let's define a pure monopoly as a single company that is "pretty much" the only company producing or selling a product (pretty much is ambiguous, but it'll work fine for our purposes. I like pohnpei's description above where we're looking at no "real" competition for a company in a given area.
Now, if a company has a threat from a substitute product, that contradicts the idea of "no real competition." So, you might be correct in some respects. For example, the Barbie brand will never likely have a monopoly over stereotypical girls' toys because of the abundance of substitutes.
However, when Standard Oil brought the Rockefeller family their fortune, there was no effective competition in terms of either substitute products, other companies in the same business, or even for other products on which consumers may have wanted to spend money! Standard Oil was a "pure monopoly" in almost any sense of the word (at its peak, it controlled 88% of the refined oil in the United States! See first link below). Because oil is a staple product, a product that is a perceived "need" for a consumer instead of an optional product, Standard Oil (before its monopoly was split up) was not really competing with other products for consumer dollars. Consumers would have inherently allocated a certain amount of money for oil.
A similar situation occurred with American Telephone and Telegraph (AT&T), which, for some time, was a government-sanctioned monopoly (see second link below). They had no problems buying up competing companies and reached the point where there was absolutely no significant competition for the telephone business. As the fastest and cheapest form of communication available for much of the century, there was no real competition for AT&T's land lines, and as a result, they could charge any price telephones' use. Again, because the telephone, with a few specific exceptions, had no real substitute and was a staple, AT&T (before its monopoly was split up) did not truly compete for consumer dollars, like Standard Oil.
Another point in talking about monopolies is the danger that they pose to the free market in terms of efficiency and pricing. Even if pure monopolies did not exist, the problems that would occur with monopolies would still exist. This effect is most efficiently seen with monopolistic pricing and oligopolies. Using game theory, it is easy to see that monopolies can adjust pricing as desired to maximize earnings (see third link). Similarly, a small group of companies can collude with each other to adjust the prices in the market to reach the monopoly price, and we have seen these "cartels" for quite some time in human history. Most recently, you see this "cartel" behavior with OPEC and NASDAQ market-making. Therefore, even if monopolies do not exist, it is important to study their theoretical structure and damage so we can deal with the problems involved in monopolistic practices performed by companies that may not be "pure monopolies."
All firms do have to compete for customer dollars, even if they have no direct competitors. For example, a firm that has a monopoly on providing cable TV in an area must compete with satellite TV and with other forms of entertainment. A firm (or governmental agency) with a monopoly on public transportation must compete to get customers to ride on public transit instead of their own private cars.
However, not all firms really face this sort of thing. For example, there is no real competition for a firm that has a monopoly over electric power in a given area. Consumers cannot realistically do without this service (as they could with cable TV or public transportation) and so these firms do not really face competition.
Therefore, I do not agree with the statement given. I do agree that pure monopolies are rare, but they do exist.
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