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 {\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 R + L {\displaystyle \mathbb {R} _{+}^{L}}; that is, the space of nonnegative real vectors of dimension L {\displaystyle L}.

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

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

D ( ⪰ p , p , e ) := { x : p x ≤ p e a n d p x ′ ≤ p e ⟹ x ′ ⪯ p x } {\displaystyle D(\succeq _{p},p,e):=\{x:p_{x}\leq p_{e}~~~and~~~p_{x'}\leq p_{e}\implies x'\preceq _{p}x\}}.

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