I remember reading somewhere that perfect competition under a free market was not really stable nor desirable.

This is because it causes profit margins to tend to 0 which has 2 negative effects:

* makes companies more fragile to externalities

* does not allow companies to accumulate treasuries that enable them to do research, innovate, take risks, etc

The end result is a stale stagnated balkanized economy. The corollary is that some market imbalances and assymetries are healthy.

Anyone knows where this reasoning comes from? Did I dream it up?

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You didn’t dream it it’s just called reality

Yes but who initially wrote it down. A lot of stuff seems obvious in hindsight but it took a bright might to notice it and formalize it.

No, that's called fiat mindset

When they tend to zero they force companies to innovate or die, it's not supposed to be stable it's supposed to evolve, there is no such thing as a final form unless a government protected monopoly exists

Exactly

Thereby disrupting "perfect competition", which as I said, is unstable. No disagreement here

This.

Reasoning doesn't seem solid.

As competition became perfectly free, and profit margins decrease, there is still difference between the capabilities of the competitors. Some companies fail first and remainders then can thrive.

Also, there are niches. You could have perfect competition between Prada and Levi to sell me jeans, and they could both still thrive.