Editing Arrow–Debreu model (section)

==== 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.