Sorry. Being a price taker means that you can’t set your own price in the market. Imagine a farmer selling wheat: the market for wheat sets the price and the farmer can either sell her wheat at that price or not, but if she asks for a higher price she will sell zilch.
Genenerally, the textbook criteria are:
- Perfect information
- Many buyers and sellers
- Ease of entry and exit to the market
- Price taking
(It’s been a long time since I did principals, so there may be another.)
So, you may have a monopolistic market that operates where others can enter the market easily, then the market may be essentially competitive because if the monopolist tries to get monopolistic profits, others will enter the market.
What’s good about the competitive market is that it is welfare maximizing; i.e., we get the maximum amount of human well-being for the amount of resources available. A monopolist will restrict supply and raise prices to get “economic profits,” which are profits over and above the accounting profits that put food on the business owner’s table.
The problem with “free markets” are that they aren’t meaningfully defined, so we have no reason to think that when one is speaking of them, they are including economic competitiveness as a necessary condition. For example, in real estate buyers and sellers have price-setting power. Because of this real estate markets are not competitive. Any person who advocates free markets for real estate is not advocating for the wellfare-maximizing benefits of competitive markets. Indeed, in some cases government regulation is necessary to make a market competitive and therefore a free market would be a bad thing. (Take a look at David Friedman’s page for an informed opinion contrary to my own on that last point.)