Allenby-Rossi (1991) Quality Perceptions and Asymmetric Switching between Brands

Summary

Allenby and Rossi introduce the nonhomothetic logit model, in which marginal utility of brand $i$ is $\phi_i(u) = \exp(a_i - k_i u)$, making indifference curves rotate rather than shift in parallel. The rotation rate $k_i$ provides a data-revealed quality ranking: small $k_i$ means marginal utility decays slowly with total utility — the brand is preferred as households become better off. Applied to 10 margarine brands (517 households, 4,470 purchase occasions, Springfield MO ERIM scanner panel), the model assigns an objective quality ordering, explains why premium brands attract disproportionately more switchers under price promotions (asymmetric switching), and achieves the best Bayesian information criterion (BIC) (−4,944.5, 17 params) and hold-out fit of all competing models — with 1/3 the parameters of nested logit.

Key Claims

• Marginal utility: $\phi_i(u) = \exp(a_i - k_i u)$; $k_i > 0$; smaller $k_i$ = higher quality (indifference curves rotate toward the quality brand as budget expands).
• At corner solution $u^i$ (consuming only brand $i$): $\ln u^i = a_i - k_i u^i + \ln(v/p_i)$; choice probabilities are standard logit in $v_i = -\ln p_i + a_i - k_i u^i$.
• Quality ranking (margarine, best to worst by $k^*$): Fleischmann's Tub ($-1.228$) > Parkay Tub ($-0.756$) > Fleischmann's Stick ($-0.619$) > … > House Tub ($1.281$).
• Quality anomaly: Imperial Stick ($k^*=0.518$, price $0.78) is lower quality than Parkay Stick ($k^*=0$, price $0.52) — quality inversion not detectable from price alone.
• Asymmetric switching: price cut on premium brand attracts more switchers because a larger proportion of households cross the utility threshold for trading up.
• Loyalty control: Guadagni-Little loyalty $L_{hj}$ = (observed − predicted frequency from first-half calibration); loyalty parameter $\delta = 0.960$ (std.err. 0.028); price sensitivity $\tau = 6.66$ (std.err. 0.379).
• Expenditure function: $\ln v = -2.12 + 0.132\ln(\text{income}) - 0.431\ln(\text{family size}) + 0.121(\text{college}) + 0.262(\text{retired}) - 2.53\ln(\text{price index})$.
• Goodness of fit ($n=4470$): Nonhomothetic BIC = −4,944.5 (17 params) vs. best competitor (nested logit, 10 prices) BIC = −5,131.7 (58 params); hold-out $\log L = -3,262.1$ vs. $-3,407.3$.
• Nested logit connection: brands with similar $k_i$ generate correlated errors in a homothetic logit, structurally mimicking nested logit groups — the paper argues much of the "correlated errors" evidence in scanner panel logit is nonhomotheticity misspecification.
• Profit maximization: nonhomothetic model yields sensible optimal prices (0.0663 → 0.0690 cents/transaction profit); standard logit produces implausible extreme prices and is useless for pricing policy.
• Note: classical maximum likelihood estimation (MLE) paper — no Bayesian Markov chain Monte Carlo (MCMC). Heterogeneity is handled through loyalty and demographics in the expenditure function, not hierarchical Bayes (HB) random effects.

Concepts Introduced or Extended

• Nonhomothetic Preferences — introduces the rotating-indifference-curve logit model; quality ranking from $k_i$ parameter; asymmetric switching mechanism
• Discrete Choice Models — nonhomothetic extension of standard logit; nested logit comparison; independence of irrelevant alternatives (IIA) and misspecification
• Market Segmentation — quality-based segmentation from $k_i$ ordering

Entities Mentioned

• Greg M. Allenby
• Peter E. Rossi

Quotes

"Price reductions in higher quality brands attract more consumers than do price reductions in lower quality brands."

"A researcher using a homothetic logit model specification would find evidence of correlated errors or departure from the IIA assumption due to model misspecification."

"The proposed model results in improved sample and predictive fits, and requires 1/3 the number of parameters as compared to standard logit and nested logit models."

"The homothetic utility specification is clearly not useful for the purpose of setting price policies."

My Take

The core theoretical insight — that rotating rather than shifting indifference curves can simultaneously explain asymmetric switching, provide an objective quality ranking, and generate apparent error correlations as a misspecification artifact — is elegant and econometrically sharp. The empirical performance is striking: 17 parameters beating 58-parameter nested logit on both in-sample BIC and out-of-sample fit. The quality anomaly on Imperial Stick (high price, low perceived quality) demonstrates genuine empirical content. The main limitation is that all heterogeneity enters through loyalty and demographics in the expenditure function — there is no continuous distribution over $k_i$ across households. A natural Bayesian extension would allow $k_i$ to vary by household, which Allenby and Rossi later pursue through HB frameworks (e.g., Allenby-Rossi 1999). The misspecification-as-nested-logit argument is the paper's most lasting methodological contribution.