Universal

Overview of Preferences and Choices

Microeconomic analysis begins with the modeling of individual decision-making, where an agent selects from a given set of feasible alternatives. The foundational framework relies on formalizing preferences over a set $X$, representing the collection of all conceivable choices or consumption bundles.

Decision theory separates the underlying preferences of an individual from their observed choices. Preferences represent subjective evaluations and rankings of alternatives, whereas choice rules describe the empirical selection of options from specific feasible sets or menus.

Preference Relations and Binary Relations

An individual's preferences over a set $X$ are formalized using a binary relation denoted by $\succeq$. The expression $x \succeq y$ is read as alternative $x$ is weakly preferred to alternative $y$, meaning that the decision-maker considers $x$ to be at least as desirable as $y$.

From the weak preference relation $\succeq$, two secondary binary relations are derived:

• Strict Preference ($\succ$): Alternative $x$ is strictly preferred to alternative $y$, denoted $x \succ y$, if and only if $x \succeq y$ and it is not the case that $y \succeq x$. This indicates that the decision-maker strictly prefers $x$ over y$.\n- Indifference (\sim$): Alternative$x$is indifferent to alternative$y$, denoted$x \sim y$, if and only if$x \succeq y$and$y \succeq x. This indicates that the decision-maker views both options as equally desirable.\n\n# Axioms of Rational Choice\n\nTo establish logical consistency and structure in decision-making, preferences are required to satisfy specific behavioral axioms. A preference relation \succeq is defined as rational if it satisfies both completeness and transitivity.\n\n- Completeness: For any two alternatives x, y \in X$, either$x \succeq y$,$y \succeq x$, or both. The completeness axiom rules out indecision, requiring that the individual is capable of comparing and ranking any pair of alternatives within the set$X$.

• Transitivity: For any three alternatives $x, y, z \in X$, if $x \succeq y$ and $y \succeq z$, then $x \succeq z$. Transitivity prevents preference cycles (such as $x \succ y$, $y \succ z$, and $z \succ x$), ensuring internal consistency in decision-making.

• Reflexivity: For any alternative $x \in X$, $x \succeq x$. Any alternative is always at least as desirable as itself. Reflexivity is implicitly satisfied under completeness.

Utility Functions and Representation

A utility function $u: X \rightarrow \mathbb{R}$ is a real-valued function that assigns a numerical value to each alternative in $X$. A utility function represents a preference relation $\succeq$ if, for all alternatives $x, y \in X$:

$x \succeq y \iff u(x) \geq u(y)$

The existence of a utility function simplifies choice analysis by transforming preference comparisons into real-number comparisons:

• Ordinal Nature: Utility functions in basic choice theory are ordinal. The specific numerical values assigned to alternatives carry no absolute intrinsic meaning; only the relative ranking of the values matters.

• Strictly Monotonic Transformations: If $u(x)$ is a utility function representing $\succeq$, and $f: \mathbb{R} \rightarrow \mathbb{R}$ is a strictly increasing function, then the composite function $v(x) = f(u(x))$ also represents the exact same preference relation $\succeq$.

• Representation Theorems: If the choice set $X$ is finite or countably infinite, any rational preference relation can be represented by a utility function. For uncountably infinite sets, such as continuous consumption spaces $\mathbb{R}_+^n$, an additional requirement of preference continuity is necessary to guarantee representation (Debreu Representation Theorem).

Choice Functions and Revealed Preference

While preference relations describe internal rankings, choice functions describe observable behavior. Let $\mathcal{B}$ be a collection of non-empty subsets of $X$, where each subset $S \in \mathcal{B}$ represents a menu of available alternatives.

• Choice Function: A choice function $C(S)$ assigns to every menu $S \in \mathcal{B}$ a non-empty subset of chosen alternatives, such that $C(S) \subseteq S$.

• Preference Rationalizability: A choice function $C(S)$ is rationalized by a binary preference relation $\succeq$ if, for every available menu $S$:

$C(S) = \{x \in S \mid x \succeq y \text{ for all } y \in S\}$

• Weak Axiom of Revealed Preference (WARP): Choice behavior satisfies WARP if, for any two alternatives $x, y \in X$ and menus $S, T \in \mathcal{B}$ containing both $x$ and $y$, if $x \in C(S)$ and $y \in C(S)$, then $y \in C(T)$ implies $x \in C(T)$. Satisfying WARP ensures that observed choice behavior can be modeled as the maximization of a rational preference relation.