Editing Arrow–Debreu model (section)

=== Notation setup ===
In general, we write indices of agents as superscripts and vector coordinate indices as subscripts.

==== useful notations for real vectors ====

* <math>x \succeq y</math> if <math>\forall n, x_n \geq y_n</math>
* <math>\R^N_+</math> is the set of <math>x</math> such that <math>x \succeq 0</math>
* <math>\R_{++}^N</math> is the set of <math>x</math> such that <math>x \succ 0</math>
* <math>\Delta_N = \left\{x\in \R^N: x_1, ..., x_N \geq 0, \sum_{n\in 1:N} x_n = 1\right\}</math> is the [[Simplex|N-simplex]]. We often call it the '''price simplex''' since we sometimes scale the price vector to lie on it.

==== market ====

* The '''commodities''' are indexed as <math>n\in 1:N</math>. Here <math>N</math> is the number of commodities in the economy. It is a finite number.
* The '''price vector''' <math>p = (p_1, ..., p_N) \in \R_{++}^N</math> is a vector of length <math>N</math>, with each coordinate being the price of a commodity. The prices may be zero or positive.

==== households ====

* The '''households''' are indexed as <math>i\in I</math>.
* Each household begins with an '''endowment''' of commodities <math>r^i\in \R^N_+</math>.
* Each household begins with a tuple of '''ownerships''' of the producers <math>\alpha^{i,j} \geq 0</math>. The ownerships satisfy <math>\sum_{i\in I} \alpha^{i,j} = 1 \quad \forall j\in J </math>.
* The budget that the household receives is the sum of its income from selling endowments at the market price, plus profits from its ownership of producers:<math display="block">M^i(p) = \langle p, r^i\rangle + \sum_{j\in J}\alpha^{i,j}\Pi^j(p)</math>(<math>M</math> stands for ''money'')
* Each household has a '''Consumption Possibility Set''' <math>CPS^i\subset \R_+^N</math>.
* Each household has a '''preference relation''' <math>\succeq^i</math> over <math>CPS^i</math>.
* With assumptions on <math>\succeq^i</math> (given in the next section), each preference relation is representable by a '''utility function''' <math>u^i: CPS^i \to [0, 1]</math> by the [[Debreu theorems]]. Thus instead of maximizing preference, we can equivalently state that the household is maximizing its utility.
* A '''consumption plan''' is a vector in <math>CPS^i</math>, written as <math>x^i</math>.
* <math>U_+^i(x^i)</math> is the set of consumption plans at least as preferable as <math>x^i</math>.
* The '''budget set''' is the set of consumption plans that it can afford:<math display="block">B^i(p) = \{x^i \in CPS^i : \langle p, x^i \rangle \leq M^i(p)\}</math>.
* For each price vector <math>p</math>, the household has a '''demand''' vector for commodities, as <math>D^i(p)\in \R_+^N</math>. This function is defined as the solution to a constraint maximization problem. It depends on both the economy and the initial distribution.<math display="block">D^i(p) := \arg\max_{x^i \in B^i(p)} u^i(x^i)</math>It may not be well-defined for all <math>p \in \R^N_{++}</math>. However, we will use enough assumptions to be well-defined at equilibrium price vectors.

==== producers ====

* The producers are indexed as <math>j\in J</math>.
* Each producer has a '''Production Possibility Set''' <math>PPS^j</math>. Note that the supply vector may have both positive and negative coordinates. For example, <math>(-1, 1, 0)</math> indicates a production plan that uses up 1 unit of commodity 1 to produce 1 unit of commodity 2.
* A '''production plan''' is a vector in <math>PPS^j</math>, written as <math>y^j</math>.
* For each price vector <math>p</math>, the producer has a '''supply''' vector for commodities, as <math>S^j(p)\in \R^N</math>. This function will be defined as the solution to a constraint maximization problem. It depends on both the economy and the initial distribution.<math display="block">S^j(p) := \arg\max_{y^j\in PPS^j} \langle p, y^j\rangle</math>It may not be well-defined for all <math>p \in \R^N_{++}</math>. However, we will use enough assumptions to be well-defined at equilibrium price vectors.
* The '''profit''' is <math display="block">\Pi^j(p) := \langle p, S^j(p)\rangle = \max_{y^j\in PPS^j} \langle p, y^j\rangle</math>

==== aggregates ====

* aggregate consumption possibility set <math>CPS = \sum_{i\in I}CPS^i</math>.
* aggregate production possibility set <math>PPS = \sum_{j\in J}PPS^j</math>.
* aggregate endowment <math>r = \sum_i r^i</math>
* aggregate demand <math>D(p) := \sum_i D^i(p)</math>
* aggregate supply <math>S(p) := \sum_j S^j(p)</math>
* excess demand <math>Z(p) = D(p) - S(p) - r</math>

==== 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.