How does Brouwer's fixed point theorem relate to Walrasian equilibrium?

Question

I am trying to understand Walrasian equilibrium and its connection to fixed points, especially how we can apply Brouwer’s fixed point theorem to the notion of Walrasian equilibrium. I understand the implications of Brouwer’s fixed point theorem, and that conversion of prices to the unit simplex is an admissible transformation such that we know, by the theorem, there must be at least one fixed point. I understand Walrasian equilibrium as a price and consumption vector such that utilities of all households are maximized s.t. budget constraints and markets clear.

What I do not understand is the link between the two. I struggle with the intuition as to how fixed points map onto equilibrium prices.

Please excuse me if I am asking a stupid / obvious question. I cannot find an explanation of the intuition of the link anywhere I have looked.

Thanks in advance for your help,

R.

Answer 1

In standard proofs of the existence of Walrasian equilibrium, after one applies the appropriate fixed point theorem (typically Kakutani's, but sometimes Brouwer's), one then shows that the given fixed point is, in fact, a Walrasian equilibrium.

Depending on the exact construction of the proof, sometimes the fixed point will only give you the equilibrium price vector. However, this will be done so that the implied demands will then clear the market.

In other proofs, like this one, the fixed point will give you both the equilibrium price and allocation.