Editing Arrow–Debreu model (section)

=== convexity vs strict convexity ===
The assumptions of strict convexity can be relaxed to convexity. This modification changes supply and demand functions from point-valued functions into set-valued functions (or "correspondences"), and the application of Brouwer's fixed-point theorem into Kakutani's fixed-point theorem.

This modification is similar to the generalization of the [[minimax theorem]] to the existence of [[Nash equilibrium|Nash equilibria]].

The two fundamental theorems of welfare economics holds without modification.
{| class="wikitable"
|+converting from strict convexity to convexity
!strictly convex case
!convex case
|-
|<math>PPS^j</math> is strictly convex
|<math>PPS^j</math> is convex
|-
|<math>CPS^i</math> is strictly convex
|<math>CPS^i</math> is convex
|-
|<math>\succeq^i</math> is strictly convex
|<math>\succeq^i</math> is convex
|-
|<math>\tilde S^j(p)</math> is point-valued
|<math>\tilde S^j(p)</math> is set-valued
|-
|<math>\tilde S^j(p)</math> is continuous
|<math>\tilde S^j(p)</math> has [[Hemicontinuity#Closed graph theorem|closed graph]] ("upper hemicontinuous")
|-
|<math>\langle p, \tilde Z(p)\rangle\leq 0</math>
|<math>\langle p, z\rangle\leq 0</math> for any <math>z\in \tilde Z(p)</math>
|-
|...
|...
|-
| equilibrium exists by Brouwer's fixed-point theorem
| equilibrium exists by Kakutani's fixed-point theorem
|}