Editing Arrow–Debreu model (section)

=== Assumptions ===
{| class="wikitable"
|+on the households
!assumption
!explanation
!can we relax it?
|-
| <math>CPS^i </math> is closed
|Technical assumption necessary for proofs to work.
|No. It is necessary for the existence of demand functions.
|-
|local nonsatiation: <math>\forall x\in CPS^i, \epsilon > 0, </math>  <math>\exists x'\in CPS^i, x' \succ^i x,\|x' - x\|< \epsilon </math>
|Households always want to consume a little more.
|No. It is necessary for Walras's law to hold.
|-
| <math>CPS^i </math> is strictly convex
|strictly [[diminishing marginal utility]]
|Yes, to mere convexity, with Kakutani's fixed-point theorem. See next section.
|-
| <math>CPS^i </math> is convex
|diminishing marginal utility
|Yes, to nonconvexity, with [[Shapley–Folkman lemma]].
|-
|continuity:  <math>U_+^i(x^i)</math> is closed.
|Technical assumption necessary for the existence of utility functions by the [[Debreu theorems]].
|No. If the preference is not continuous, then the excess demand function may not be continuous.
|-
| <math>U_+^i(x^i)</math> is strictly convex.
|For two consumption bundles, any bundle between them is better than the lesser.
|Yes, to mere convexity, with Kakutani's fixed-point theorem. See the next section.
|-
| <math>U_+^i(x^i)</math> is convex.
|For two consumption bundles, any bundle between them is no worse than the lesser.
|Yes, to nonconvexity, with [[Shapley–Folkman lemma]].
|-
|The household always has at least one feasible consumption plan.
|no bankruptcy
|No. It is necessary for the existence of demand functions.
|}
{| class="wikitable"
|+on the producers
!assumption
!explanation
!can we relax it?
|-
| <math>PPS^j </math> is strictly convex
|[[diseconomies of scale]]
|Yes, to mere convexity, with Kakutani's fixed-point theorem. See next section.
|-
| <math>PPS^j </math> is convex
|no [[economies of scale]]
|Yes, to nonconvexity, with [[Shapley–Folkman lemma]].
|-
| <math>PPS^j </math> contains 0.
|Producers can close down for free.
|
|-
| <math>PPS^j </math> is a closed set
|Technical assumption necessary for proofs to work.
|No. It is necessary for the existence of supply functions.
|-
| <math>PPS \cap \R_+^N </math> is bounded
|There is no arbitrarily large "free lunch".
|No. Economy needs scarcity.
|-
| <math>PPS \cap (-PPS) </math> is bounded
|The economy cannot reverse arbitrarily large transformations.
|
|}