# Demand set

A demand set is a model of the most-preferred bundle of goods an agent can afford. The set is a function of the preference relation for this agent, the prices of goods, and the agent's endowment.

Assuming the agent cannot have a negative quantity of any good, the demand set can be characterized this way:

Define ${\displaystyle L}$ as the number of goods the agent might receive an allocation of. An allocation to the agent is an element of the space ${\displaystyle \mathbb {R} _{+}^{L}}$; that is, the space of nonnegative real vectors of dimension ${\displaystyle L}$.

Define ${\displaystyle \succeq _{p}}$ as a weak preference relation over goods; that is, ${\displaystyle x\succeq _{p}x'}$ states that the allocation vector ${\displaystyle x}$ is weakly preferred to ${\displaystyle x'}$.

Let ${\displaystyle e}$ be a vector representing the quantities of the agent's endowment of each possible good, and ${\displaystyle p}$ be a vector of prices for those goods. Let ${\displaystyle D(\succeq _{p},p,e)}$ denote the demand set. Then:

${\displaystyle D(\succeq _{p},p,e):=\{x:p_{x}\leq p_{e}~~~and~~~p_{x'}\leq p_{e}\implies x'\preceq _{p}x\}}$.