Editing Arrow–Debreu model (section)

==== 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>