Editing Arrow–Debreu model (section)

==== the whole economy ====

* An '''economy''' is a tuple <math>(N, I, J, CPS^i, \succeq^i, PPS^j)</math>. It is a tuple specifying the commodities, consumer preferences, consumption possibility sets, and producers' production possibility sets.
* An '''economy with initial distribution''' is an economy, along with an initial distribution tuple <math>(r^i, \alpha^{i,j})_{i\in I, j\in J}</math> for the economy.
* A '''state''' of the economy is a tuple of price, consumption plans, and production plans for each household and producer: <math>((p_n)_{n\in 1:N}, (x^i)_{i\in I}, (y^j)_{j\in J})</math>.
* A state is '''feasible''' iff each <math>x^i \in CPS^i</math>, each <math>y^j\in PPS^j</math>, and <math>\sum_{i\in I}x^i \preceq \sum_{j\in J}y^j + r</math>.
* The feasible production possibilities set, given endowment <math>r</math>, is <math>PPS_r := \{y\in PPS: y+r \succeq 0\}</math>.
* Given an economy with distribution, the '''state corresponding to a price vector''' <math>p</math> is <math>(p, (D^i(p))_{i\in I}, (S^j(p))_{j\in J})</math>.
* Given an economy with distribution, a price vector <math>p</math> is an '''equilibrium price vector''' for the economy with initial distribution, iff<math display="block">Z(p)_n \begin{cases}
	  \leq 0 \text{ if } p_n = 0 \\
	  = 0 \text{ if } p_n > 0 
	  \end{cases}</math>That is, if a commodity is not free, then supply exactly equals demand, and if a commodity is free, then supply is equal or greater than demand (we allow free commodity to be oversupplied).
* A state is an '''equilibrium state''' iff it is the state corresponding to an equilibrium price vector.