Nonhomothetic Preferences

Definition

A preference ordering is nonhomothetic if the optimal consumption bundle changes along any budget ray when income rises — equivalently, if indifference curves are not radially symmetric (they rotate rather than shift in parallel). In discrete choice models, nonhomotheticity means that brand-choice probabilities depend on the level of product-class expenditure, not just relative prices. This generates income effects within the product category and is the theoretical basis for asymmetric switching between quality tiers.

Key Ideas

Rotating indifference curves: Standard logit uses utility $u(x) = \xi'x$ with constant marginal utilities; nonhomothetic logit uses $\phi_i(u) = \exp(a_i - k_i u)$ so that marginal utility falls at rate $k_i$ as total utility $u$ rises. Indifference curves rotate toward the low- $k_i$ brand as expenditure increases.
Quality ranking from $k_i$ : Smaller $k_i$ = higher quality. As households allocate more expenditure to the product class (due to income growth or a price promotion on any brand), they rotate toward low- $k_i$ brands. No attribute data or price-quality assumptions required; quality is identified from revealed-preference rotation.
Asymmetric switching: A price cut on a premium brand (low $k_i$ ) attracts more switchers than an equal-size cut on an economy brand (high $k_j$ ), because it brings a larger share of households above the utility threshold for trading up.
Corner solution and utility equation: At the corner solution consuming only brand $i$ , utility $u^i$ solves the fixed-point equation $\ln u^i = a_i - k_i u^i + \ln(v/p_i)$ , where $v$ is product-class expenditure and $p_i$ is price. Newton's method always converges to a positive solution.
Choice probabilities: With iid extreme value errors, $\Pr(i) = \exp(\tau v_i)/\sum_j \exp(\tau v_j)$ where $v_i = -\ln p_i + a_i - k_i u^i$ — standard logit structure in the modified utility index $v_i$ .

How It Works

Expenditure allocation (two-stage budgeting)

Weak separability: $U(x,z) = U(u(x),h(z))$ . In stage 1, household allocates $v$ to the product class via an expenditure function with demographics and price index. In stage 2, the choice among brands is made by maximizing $u(x)$ subject to $p'x \leq v$ .

Nonhomothetic choice probabilities (Allenby-Rossi 1991)

For each brand $i$ , solve $\ln u^i = a_i - k_i u^i + \ln(v/p_i)$ for $u^i$ (Newton iteration).
Compute $v_i = -\ln p_i + a_i - k_i u^i$ .
$\Pr(i) = \exp(\tau v_i)/\sum_j \exp(\tau v_j)$ .

Nesting without explicit nesting

Brands with similar $k_i$ values generate positively correlated error terms under a homothetic logit — because price shocks that increase $u$ rotate all household preferences toward low- $k$ brands simultaneously. This structural correlation mimics nested logit without explicit group definitions. The paper argues that much of the "correlated errors" evidence in scanner panel logit fitting is nonhomotheticity misspecification.

Why It Matters

Resolves the asymmetric-switching puzzle (premium-brand promotions out-pull economy-brand promotions) without inventing additional parameters or arbitrary nesting structures.
Provides an objective, attribute-free quality measure $k_i$ from revealed preference: quality is inferred from the direction of rotation in choice probabilities as expenditure varies.
Achieves parsimony: in the margarine application, 17 parameters beat 58-parameter nested logit on both in-sample Bayesian Information Criterion (BIC) and out-of-sample predictive likelihood.
Produces sensible profit-maximizing prices; homothetic logit collapses under pricing optimization.
Suggests that nested logit's apparent empirical successes may be partly artifacts of utility misspecification.

Open Questions

Heterogeneous $k_i$ : allowing quality rotation rates to vary by household (Hierarchical Bayes (HB) random effects over $k_i$ ) would capture the interaction between quality perception and income heterogeneity; Allenby-Rossi pursue this direction in later Bayesian work.
Continuous purchase quantities: Hanemann (1984) discrete/continuous models are the natural complement when quantities, not just brand choice, are the outcome.
Multi-category extension: does the quality rotation mechanism interact with complementarity across product categories?