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...
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