Editing Arrow–Debreu model (section)

=== Accounting for strategic bargaining ===
In the model, all producers and households are "[[Market power|price takers]]", meaning that they transact with the market using the price vector <math>p</math>. In particular, behaviors such as cartel, monopoly, consumer coalition, etc are not modelled. [[Edgeworth's limit theorem]] shows that under certain stronger assumptions, the households can do no better than price-take at the limit of an infinitely large economy.

==== Setup ====
In detail, we continue with the economic model on the households and producers, but we consider a different method to design production and distribution of commodities than the market economy. It may be interpreted as a model of a "socialist" economy.

* There is no money, market, or private ownership of producers.
* Since we have abolished private ownership, money, and the profit motive, there is no point in distinguishing one producer from the next. Consequently, instead of each producer planning individually <math>y^j \in PPS^j</math>, it is as if the whole society has one great producer producing <math>y\in PPS</math>.
* Households still have the same preferences and endowments, but they no longer have budgets.
* Producers do not produce to maximize profit, since there is no profit. All households come together to make a '''state''' <math>((x_i)_{i\in I}, y)</math>—a production and consumption plan for the whole economy—with the following constraints:<math display="block">x^i \in CPS^i, y \in PPS, y\succeq \sum_i (x^i- r^i)</math>
* Any nonempty subset of households may eliminate all other households, while retaining control of the producers.

This economy is thus a [[Cooperative game theory|cooperative game]] with each household being a player, and we have the following concepts from cooperative game theory:

* A '''blocking coalition''' is a nonempty subset of households, such that there exists a strictly Pareto-better plan even if they eliminate all other households.
* A state is a '''core state''' iff there are no blocking coalitions.
* The '''core of an economy''' is the set of core states.
Since we assumed that any nonempty subset of households may eliminate all other households, while retaining control of the producers, the only states that can be executed are the core states. A state that is not a core state would immediately be objected by a coalition of households.

We need one more assumption on <math>PPS</math>, that it is a '''cone''', that is, <math>k \cdot PPS \subset PPS</math> for any <math>k \geq 0</math>. This assumption rules out two ways for the economy to become trivial.

* The curse of free lunch: In this model, the whole <math>PPS</math> is available to any nonempty coalition, even a coalition of one. Consequently, if nobody has any endowment, and yet <math>PPS</math> contains some "free lunch" <math>y\succ 0</math>, then (assuming preferences are monotonic) every household would like to take all of <math>y</math> for itself, and consequently there exists *no* core state. Intuitively, the picture of the world is a committee of selfish people, vetoing any plan that doesn't give the entire free lunch to itself.
* The limit to growth: Consider a society with 2 commodities. One is "labor" and another is "food". Households have only labor as endowment, but they only consume food. The <math>PPS</math> looks like a ramp with a flat top. So, putting in 0-1 thousand hours of labor produces 0-1 thousand kg of food, linearly, but any more labor produces no food. Now suppose each household is endowed with 1 thousand hours of labor. It's clear that every household would immediately block every other household, since it's always better for one to use the entire <math>PPS</math> for itself.

==== 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.