Editing Arrow–Debreu model (section)

=== existence of general equilibrium ===
As the last piece of the construction, we define '''[[Walras's law]]''':

* The unrestricted market satisfies Walras's law at <math>p</math> iff all <math>S^j(p), D^i(p)</math> are defined, and <math>  \langle p, Z(p)\rangle = 0</math>, that is,<math display="block"> \sum_{j\in J} \langle p,S^j(p)\rangle + \langle p, r\rangle = \sum_{i\in I} \langle p, D^i(p)\rangle</math>
* The restricted market satisfies Walras's law at <math>p</math> iff <math>  \langle p, \tilde Z(p)\rangle = 0</math>.

Walras's law can be interpreted on both sides:

* On the side of the households, it is said that the aggregate household expenditure is equal to aggregate profit and aggregate income from selling endowments. In other words, every household spends its entire budget.
* On the side of the producers, it is saying that the aggregate profit plus the aggregate cost equals the aggregate revenue.
{{Math theorem
| name = Theorem
| note = 
| math_statement = <math>\tilde Z</math> satisfies ''weak'' Walras's law: For all <math>p \in \R_{++}^N</math>, 
<math display="block">\langle p, \tilde Z(p)\rangle \leq 0</math>
and if <math>\langle p, \tilde Z(p)\rangle < 0</math>, then <math>\tilde Z(p)_n > 0</math> for some <math>n</math>.
}}

{{Math proof|title=Proof sketch|proof=  
If total excess demand value is exactly zero, then every household has spent all their budget. Else, some household is restricted to spend only part of their budget. Therefore, that household's consumption bundle is on the boundary of the restriction, that is, <math>\|\tilde D^i(p)\| = C</math>. We have chosen (in the previous section) <math>C</math> to be so large that even if all the producers coordinate, they would still fall short of meeting the demand. Consequently there exists some commodity <math>n</math> such that <math>\tilde D^i(p)_n > \tilde S(p)_n + r_n</math> 
}}

{{Math theorem
| name = Theorem
| note = 
| math_statement = An equilibrium price vector exists for the restricted market, at which point the restricted market satisfies Walras's law.
}}

{{Math proof|title=Proof sketch|proof=  
 By definition of equilibrium, if <math>p</math> is an equilibrium price vector for the restricted market, then at that point, the restricted market satisfies Walras's law.

<math>\tilde Z</math> is continuous since all <math>\tilde S^j, \tilde D^i</math> are continuous.

Define a function <math display="block">f(p) = \frac{\max(0, p + \gamma \tilde Z(p))}{\sum_n \max(0, p_n + \gamma \tilde Z(p)_n)}</math>on the price simplex, where <math>\gamma</math> is a fixed positive constant.

By the weak Walras law, this function is well-defined. By Brouwer's fixed-point theorem, it has a fixed point. By the weak Walras law, this fixed point is a market equilibrium.
}}

Note that the above proof does not give an iterative algorithm for finding any equilibrium, as there is no guarantee that the function <math>f</math> is a [[Contraction mapping|contraction]]. This is unsurprising, as there is no guarantee (without further assumptions) that any market equilibrium is a stable equilibrium.

{{Math theorem
| name = Corollary
| note = 
| math_statement = An equilibrium price vector exists for the unrestricted market, at which point the unrestricted market satisfies Walras's law.
}}