Editing Arrow–Debreu model (section)

==== Main results (Debreu and Scarf, 1963) ====
{{Math theorem|name=Proposition|math_statement= Market equilibria are core states.
}}

{{Math proof|title=Proof|proof=

Define the price hyperplane <math>\langle p, q \rangle = \langle p, \sum_j y^j\rangle</math>. Since it's a supporting hyperplane of <math>PPS</math>, and <math>PPS</math> is a convex cone, the price hyperplane passes the origin. Thus <math>\langle p, \sum_j y^j\rangle =  \langle p, \sum_i x^i - r^i\rangle = 0</math>.

Since <math>\sum_j \langle p, y^j\rangle</math> is the total profit, and every producer can at least make zero profit (that is, <math>0 \in PPS^j</math> ), this means that the profit is exactly zero for every producer. Consequently, every household's budget is exactly from selling endowment.

<math display="block">\langle p, x^i \rangle = \langle p, r^i\rangle</math>

By utility maximization, every household is already doing as much as it could. Consequently, we have <math>\langle p, U^i_{++}(x^i)\rangle > \langle p, r^i\rangle</math>.

In particular, for any coalition <math>I' \subset I</math>, and any production plan <math>x'^i</math> that is Pareto-better, we have

<math display="block"> \sum_{i\in I'} \langle p, x'^i \rangle >\sum_{i\in I'} \langle p, r^i \rangle</math>
and consequently, the point <math>\sum_{i\in I'} x'^i - r^i</math> lies above the price hyperplane, making it unattainable.
}}

In Debreu and Scarf's paper, they defined a particular way to approach an infinitely large economy, by "replicating households". That is, for any positive integer <math>K</math>, define an economy where there are <math>K</math> households that have exactly the same consumption possibility set and preference as household <math>i</math>.

Let <math>x^{i, k}</math> stand for the consumption plan of the <math>k</math>-th replicate of household <math>i</math>. Define a plan to be '''equitable''' iff <math>x^{i, k} \sim^i x^{i, k'}</math> for any <math>i\in I</math> and <math>k, k'\in K</math>.

In general, a state would be quite complex, treating each replicate differently. However, core states are significantly simpler: they are equitable, treating every replicate equally.

{{Math theorem|name=Proposition|math_statement= Any core state is equitable.}}

{{Math proof|title=Proof|proof=  We use the "underdog coalition".

Consider a core state <math>x^{i, k}</math>. Define average distributions <math>\bar x^{i} := \frac 1K \sum_{k\in K} x^{i,k}</math>.

It is attainable, so we have <math>K \sum_{i} (\bar x^i - r^i) \in PPS</math>

Suppose there exist any inequality, that is, some <math>x^{i, k} \succ^i x^{i, k'}</math>, then by convexity of preferences, we have <math>\bar x^i \succ^i x^{i, k'}</math>, where <math>k'</math> is the worst-treated household of type <math>i</math>.

Now define the "underdog coalition" consisting of the worst-treated household of each type, and they propose to distribute according to <math>\bar x^i</math>. This is Pareto-better for the coalition, and since <math>PP</math> is conic, we also have <math>\sum_i(\bar x^i - r^i) \in PPS</math>, so the plan is attainable. Contradiction.
}}

Consequently, when studying core states, it is sufficient to consider one consumption plan for each type of households. Now, define <math>C_K</math> to be the set of all core states for the economy with <math>K</math> replicates per household. It is clear that <math>C_1 \supset C_2 \supset \cdots</math>, so we may define the limit set of core states <math>C := \cap_{K=1}^\infty C_K</math>.

We have seen that <math>C</math> contains the set of market equilibria for the original economy. The converse is true under minor additional assumption:<ref>(Starr 2011) Theorem 22.2</ref>

{{Math theorem
| name = (Debreu and Scarf, 1963)
| note = 
| math_statement = If <math>PPS</math> is a polygonal cone, or if every <math>CPS^i</math> has nonempty interior with respect to <math>\R^N</math>, then <math>C</math> is the set of market equilibria for the original economy.
}}

The assumption that <math>PPS</math> is a polygonal cone, or every <math>CPS^i</math> has nonempty interior, is necessary to avoid the technical issue of "quasi-equilibrium". Without the assumption, we can only prove that <math>C</math> is contained in the set of quasi-equilibria.