Owning the market would conceivably allow the producer to maximize producer surplus and minimize consumer surplus. In other words, charge the highest possible prices consumers are willing to pay. (As stated above, they can't charge more than consumers are willing to pay.)

However, doing so would send a strong price signal to entrepreneurs, telling them that there is an excellent opportunity at hand, namely to develop a substitute or competing product.

In other words, maximizing prices invites all sorts of competition.

This is why so-called exploitative monopolies cannot endure long term, unless they have government regulations in place to keep would-be competitors from the entering the marketplace.