Editing Arrow–Debreu model (section)

=== Number of equilibria ===
{{See also|General equilibrium theory#Uniqueness}}
Certain economies at certain endowment vectors may have infinitely equilibrium price vectors. However, "generically", an economy has only finitely many equilibrium price vectors. Here, "generically" means "on all points, except a closed set of Lebesgue measure zero", as in [[Sard's theorem]].<ref>{{Citation |last=Debreu |first=Gérard |title=Stephen Smale and the Economic Theory of General Equilibrium |date=June 2000 |url=http://dx.doi.org/10.1142/9789812792815_0025 |work=The Collected Papers of Stephen Smale |pages=243–258 |publisher=World Scientific Publishing Company |doi=10.1142/9789812792815_0025 |isbn=978-981-02-4991-5 |access-date=2023-01-06}}</ref><ref>{{Citation |last=Smale |first=Steve |title=Chapter 8 Global analysis and economics |date=1981-01-01 |url=https://www.sciencedirect.com/science/article/pii/S1573438281010126 |series=Handbook of Mathematical Economics |volume=1 |pages=331–370 |publisher=Elsevier |doi=10.1016/S1573-4382(81)01012-6 |isbn=978-0-444-86126-9 |language=en |access-date=2023-01-06}}</ref>

There are many such genericity theorems. One example is the following:<ref>{{Cite journal |last=Debreu |first=Gérard |date=December 1984 |title=Economic Theory in the Mathematical Mode |url=http://dx.doi.org/10.2307/3439651 |journal=The Scandinavian Journal of Economics |volume=86 |issue=4 |pages=393–410 |doi=10.2307/3439651 |jstor=3439651 |issn=0347-0520}}</ref><ref>(Starr 2011) Section 26.3</ref>

{{Math theorem
| name = Genericity
| math_statement = For any strictly positive endowment distribution <math>r^1, ..., r^I \in \R_{++}^N</math>, and any strictly positive price vector <math>p\in \R_{++}^N</math>, define the excess demand <math>Z(p, r^1, ..., r^I)</math> as before.

If on all <math>p, r^1, ..., r^I \in \R_{++}^N</math>,
* <math>Z(p, r^1, ..., r^I)</math> is well-defined,
* <math>Z</math> is differentiable,
* <math>\nabla_p Z</math> has <math>(N-1)</math>,

then for generically any endowment distribution <math>r^1, ..., r^I \in \R_{++}^N</math>, there are only finitely many equilibria <math>p^* \in \R_{++}^N</math>.
}}

{{Math proof|title=Proof (sketch)|proof=

Define the "equilibrium manifold" as the set of solutions to <math>Z=0</math>. By Walras's law, one of the constraints is redundant. By assumptions that <math>\nabla_p Z</math> has rank <math>(N-1)</math>, no more constraints are redundant. Thus the equilibrium manifold has dimension <math>N \times I</math>, which is equal to the space of all distributions of strictly positive endowments <math>\R_{++}^{N \times I}</math>.

By continuity of <math>Z</math>, the projection is closed. Thus by Sard's theorem, the projection from the equilibrium manifold to <math>\R_{++}^{N \times I}</math> is critical on only a set of measure 0. It remains to check that the preimage of the projection is generically not just discrete, but also finite.
}}