Structural Preference Analysis and Utility Recovery

This section delineates axiomatic tests for specific preference structures, including homotheticity, quasilinearity, and separability, as well as the formal methodology for utility recovery.

Homothetic Preferences (HARP)

Reference Implementation: validate_proportional_scaling(log)

The Homothetic Axiom of Revealed Preference (HARP) evaluates whether an agent’s preferences are invariant to proportional scaling of income, implying that commodity demand scales linearly with total expenditure.

Formal Definition:

Define the expenditure ratio \(r_{ij}\) as the cost of bundle \(j\) evaluated at prices \(i\) relative to the actual expenditure at observation \(i\):

\[r_{ij} = \frac{p^i \cdot x^i}{p^i \cdot x^j}\]

The HARP Condition:

\[\text{HARP is satisfied} \iff \nexists \text{ cycle } i_1 \to i_2 \to \cdots \to i_m \to i_1 : \prod_{k=1}^{m} r_{i_k, i_{k+1}} > 1\]

Equivalently, in logarithmic space:

\[\sum_{k=1}^{m} \log r_{i_k, i_{k+1}} \leq 0\]

Reference: Varian (1983).

Quasilinear Utility (Income Invariance)

Reference Implementation: test_income_invariance(log)

Quasilinearity implies a utility function of the form \(U(x, m) = v(x) + m\), where the demand for commodity \(x\) is independent of the agent’s income level \(m\). This is evaluated via the condition of cyclic monotonicity.

The Quasilinearity Condition:

For any sequence of observations forming a cycle \(i_1 \to i_2 \to \cdots \to i_m \to i_1\), the following must hold:

\[\sum_{k=1}^{m} p^{i_k} \cdot (x^{i_{k+1}} - x^{i_k}) \geq 0\]

Behavioral Interpretation:

A failure of quasilinearity suggests that the agent’s marginal utility of income is not constant, and choices are influenced by income effects rather than relative prices alone.

Reference: Rochet (1987).

Weak Separability (Feature Independence)

Reference Implementation: test_feature_independence(log, group_a, group_b)

Weak separability posits that preferences over a subset of commodities (Group A) are independent of the consumption levels of another subset (Group B). Formally, \(U(x_A, x_B) = V(u_A(x_A), u_B(x_B))\).

Analytical Heuristic:

The implementation evaluates separability by examining the consistency (CCEI) of choices within partitioned commodity groups and assessing the degree of cross-group correlation.

Reference: Chambers & Echenique (2016).

Utility Recovery via Afriat’s Inequalities

Reference Implementation: fit_latent_values(log)

If the observed data satisfy GARP, Afriat’s Theorem guarantees the existence of a continuous, monotonic, and concave utility function that rationalizes the behavior. PrefGraph recovers the latent utility values \(U_k\) and marginal utilities of income (Lagrange multipliers) \(\lambda_k > 0\).

Linear Programming Formulation:

The recovery is achieved by solving a system of Afriat inequalities for all observation pairs \((k, l)\):

\[U_k \leq U_l + \lambda_l \cdot p^l \cdot (x^k - x^l) \quad \forall \, k, l\]

Optimization Objective:

\[\min \sum_{k=1}^{T} \lambda_k\]

The resulting utility function is the lower envelope of the recovered tangent planes, providing a piecewise linear and concave approximation of the agent’s true preferences.

References: Afriat (1967), Varian (1982).