title: ## Contents
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debugInConsole: false # Print debug info in Obsidian console1 Preliminary Material
The term preference relation is used to refer to orderings that describe human preferences for one thing over an other.
2 Consumer Theory
Consumer theory is a branch of microeconomics that studies how people decide what to spend their money on based on their preferences and budget constraints.
A consumer is a pair \((X,\succeq)\) consisting of consumption space \(X\) and a preference relation \(\succeq\) on \(X\).
The set \(X\) represents all the possible consumption bundles that the consumer can choose. The preference relation \(\succeq\) represents a description of the consumers choice behavior (i.e. specifies what the consumer chooses from each pair of alternative bundles in \(X\)). We assume throughout that \(X=\mathbb{R}_{+}^L\) where the consumer can choose any non-negative continuous quantity of \(L\) goods.
Letting \(p \in X\) and \(W\geq 0\) the budget set for a consumer when prices are \(p\) and income is \(W\) is \[ B(p,W):=\{ y \in \mathbb{R}_{+}^L:p \cdot y \leq W \}. \]
Given a preference relation \(\succeq\), the optimal choices of the consumer are \[ x^*(p,W)=\{ x \in B(p,W):\forall y \in B(p,W),~x \succeq y \}. \] The mapping from \((p,W)\) into \(x^*(p,W)\) is the consumer’s demand correspondence. The theory of the consumer predicts that demand (choices made from budget sets) are optimal choices according to an underlying (rational) preference relation.