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

Define $$\succeq_p$$ as a weak preference relation over goods; that is, $$x \succeq_p x'$$ states that the allocation vector $$x$$ is weakly preferred to $$x'$$.

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 $$D(\succeq_p,p,e)$$ denote the demand set. Then:

$$D(\succeq_p,p,e) := \{x: p_x \leq p_e and p_{x'}\leq p_e \implies x'\preceq_p x   \}$$.