Suppose the firm tried to charge a price higher than P*, for example PH. If it did it would only sell QU, and we know that QU isn't profit maximizing.

    If the firm tries to lower the price below P*, say to PL it will sell QO units, which we have already seen is not profit maximizing. So, while it is true that a monopolist can charge any price it wants, it can't do so if it wants to maximize profits. A movie theater can charge $400 for a box of popcorn (seems like almost that now doesn't it?) but they probably won't sell any at that price.

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