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

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

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

$D(\backslash succeq\_p,p,e)\; :=\; \backslash \{x:\; p\_x\; \backslash leq\; p\_e\; ~~~and~~~\; p\_\{x\text{'}\}\backslash leq\; p\_e\; \backslash implies\; x\text{'}\backslash preceq\_p\; x\; \backslash \}$.