Theoretical Foundations#
This chapter delineates the formal notation, foundational assumptions, and axiomatic hierarchy that constitute the theoretical core of revealed preference analysis.
Formal Notation#
The analysis of choice behavior in PrefGraph is based on the following mathematical conventions:
\(p^t \in \mathbb{R}^n_+\) |
Price vector associated with observation \(t\). |
\(x^t \in \mathbb{R}^n_+\) |
Commodity bundle (quantity vector) selected at observation \(t\). |
\(e_t = p^t \cdot x^t\) |
Total expenditure at observation \(t\). |
\(E_{ij} = p^i \cdot x^j\) |
The hypothetical cost of bundle \(j\) evaluated at the price vector of observation \(i\). |
\(T\) |
Total number of longitudinal observations for a given agent. |
\(n\) |
Dimensionality of the commodity space (number of distinct goods). |
Maintained Assumptions#
The validity of revealed preference results is contingent upon several maintained assumptions regarding the underlying data-generating process. Violations of these assumptions may lead to spurious detections of behavioral inconsistency.
Assumption |
Implications of Violation |
|
|---|---|---|
A1 |
Preference Stability - The agent possesses a time-invariant utility function \(U(x)\) across all observations. |
Evolutionary changes in preferences (e.g., taste formation) may be incorrectly identified as axiomatic violations. |
A2 |
Utility Maximization - Observed choices represent the solution to \(\arg\max_x U(x)\) subject to the budget constraint. |
Decision heuristics, cognitive load, or satisficing behavior generate violations that reflect bounded rationality. |
A3 |
Local Non-Satiation - The agent strictly prefers more of at least one good; the entire budget is exhausted. |
While free disposal is mathematically accommodated, systematic under-spending or unobserved saving violates the budget model. |
A4 |
Unitary Decision-Maker - Observed choices reflect the preferences of a single optimizing agent. |
Aggregated household data or multi-user accounts may exhibit violations arising from collective choice dynamics. |
A5 |
Information Completeness - The analyst observes the exhaustive set of commodities and prices relevant to the agent’s decision. |
Partial observation (e.g., omitting essential categories) may result in an incomplete budget set, leading to false-positive violations. |
Axiomatic Hierarchy#
The fundamental axioms of revealed preference exhibit a nested hierarchical structure, providing varying levels of stringency for behavioral analysis.
Logical Relationship Between Axioms
SARP \(\Rightarrow\) GARP \(\Rightarrow\) WARP
WARP (Weak Axiom): Precludes direct contradictions in pairwise choices (cycles of length 2).
GARP (Generalized Axiom): Precludes transitive contradictions across cycles of any length, provided at least one preference is strict.
SARP (Strong Axiom): Precludes all preference cycles, including those involving indifference; it is the most restrictive condition.
Empirical applications typically focus on GARP, as it provides the necessary and sufficient conditions for rationalizability by a continuous, monotonic, and concave utility function (Afriat, 1967).
Note
Axiomatic Selection Criteria:
WARP: Employed as a computationally efficient preliminary filter for direct inconsistencies.
GARP: The standard benchmark for consumer rationality and utility maximization.
SARP: Required for applications necessitating unique demand systems or differentiable utility specifications.