Editing Arrow–Debreu model (section)

==== 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.