Identification and Estimation of Consumers' Preferences from Repeated Observations under Nonlinear Pricing

Abstract: We develop a nonparametric approach to identify and estimate consumer preferences and unobserved heterogeneity under nonlinear price schedules. Leveraging variation across multiple price schedules, we show that both the utility function and the distribution of preference types can be nonparametrically identified. The quantile function of unobserved types becomes solution of a functional equation, and we derive conditions ensuring identification. We propose an iterative approach for estimation, in which the regularization bias decays exponentially in the number of iterations while the variance grows only polynomially, yielding a near-parametric convergence rate. We propose a valid bootstrap procedure for finite-sample inference and extend the framework to accommodate potential endogeneity of prices and additional observed heterogeneity. Monte Carlo simulations and an empirical application to data from a European mail carrier demonstrate how we can recover the utility functions and preference distributions in finite samples.

• The paper proposes a novel nonparametric framework for separately identifying utility functions and unobserved consumer heterogeneity via variations in nonlinear price schedules.
• It employs iterative functional equations and regularized series solutions to achieve near-parametric convergence rates while addressing endogeneity and covariate effects.
• Empirical applications and simulations demonstrate robust finite-sample performance and enable counterfactual elasticity analysis under policy-induced price variations.

Nonparametric Identification and Estimation of Consumer Preferences under Nonlinear Pricing

Overview and Contributions

This paper develops a nonparametric framework for the identification and estimation of consumer preferences and unobserved heterogeneity from repeated cross-sectional data where consumers face nonlinear price schedules. The central innovation is exploiting variation across multiple price regimes (distinct nonlinear price schedules) to achieve separate identification of the utility function and the distribution of unobservable preference types, while imposing minimal functional form restrictions. The identification is realized through a novel linear iterative functional equation linking the quantile function of unobserved heterogeneity to the differences in marginal prices induced by the price schedule changes. Estimation proceeds via an explicit regularized iterative series solution. The methodology further accommodates endogenous prices and observed covariates and demonstrates near-parametric rates of convergence under regularity.

Model Structure

The analysis considers consumers who choose a quantity $Q$ of a good offered under a nonlinear price schedule $P(Q)$. Each consumer’s utility is of the form $u(Q) + Q\varepsilon$, with $u(\cdot)$ common across consumers and $\varepsilon$ capturing unobserved, continuously distributed preference heterogeneity. The consumer's observed quantity choice solves

$\max_Q u(Q) + Q\varepsilon - \tau(P(Q) - B)$

where $\tau > 0$ is a Lagrange multiplier, and $B$ the budget (not observed, always binding). The first-order condition links $u'(\cdot)$, $P'(\cdot)$, and $P(Q)$0:

$P(Q)$1

With only one observed schedule, $P(Q)$2 are not separately identified from the observed cross-sectional quantity distribution.

The crux of the framework is to leverage two (or more) distinct price schedules, $P(Q)$3 and $P(Q)$4, with associated observed distributions of $P(Q)$5, denoted $P(Q)$6 and $P(Q)$7. The observable cumulative distributions for each schedule satisfy

$P(Q)$8

Identification via Iterative Functional Equation

By taking differences between the two schedules at quantile-matched quantities, the authors establish the core functional equation for the quantile function $P(Q)$9, for $u(Q) + Q\varepsilon$0:

$u(Q) + Q\varepsilon$1

or, after normalization and transformation,

$u(Q) + Q\varepsilon$2

where $u(Q) + Q\varepsilon$3, and $u(Q) + Q\varepsilon$4 is the difference in marginal prices mapped at the percentile. This linear functional equation defines a unique $u(Q) + Q\varepsilon$5 (up to location, removed by normalization), under continuity and regularity of the quantile function.

The solution is shown to possess the explicit series representation:

$u(Q) + Q\varepsilon$6

with convergence under mild contraction and Hölder continuity conditions.

Figure 1: Estimation of the quantile function of $u(Q) + Q\varepsilon$7 (left) and the utility function $u(Q) + Q\varepsilon$8 (right) for simulated samples (Design 1, $u(Q) + Q\varepsilon$9).

Estimation and Asymptotics

Estimation proceeds by replacing population objects with their kernel-based empirical counterparts, applying the iterative formula for a finite number of steps (regularization parameter $u(\cdot)$0). Statistical properties are rigorously analyzed: the bias declines exponentially in $u(\cdot)$1, while the variance grows only polynomially, enabling choice of $u(\cdot)$2 for optimal bias-variance trade-off. The resulting estimator achieves nearly $u(\cdot)$3 convergence (up to log factors) for the entire function, outperforming standard regularization approaches in ill-posed inverse problems.

A multiplier bootstrap (empirical process–based) is provided for valid uniform inference. Simulation experiments confirm that the iterative estimator has strong finite-sample performance, with accuracy improving rapidly with sample size, and compare favorably against Tikhonov regularization.

Figure 2: Estimation of the quantile function of $u(\cdot)$4 (left) and the utility function $u(\cdot)$5 (right) under a crossing-support design (Design 2, $u(\cdot)$6).

Figure 3: Bootstrap confidence intervals for quantile function and $u(\cdot)$7 (Design 1, $u(\cdot)$8); Monte Carlo and bootstrap intervals compared.

Empirical Application

The model is applied to administrative data from a European mail carrier’s unaddressed advertising contracts. The estimation is performed at the sector level for two cross-sections (2009 and 2012), under the assumption of constant underlying consumer preference distribution across years. The inverse demand (price schedule) function is first estimated using an instrumental variables procedure (leveraging observable covariates assumed exogenous to pricing), and then the nonparametric estimation approach is applied.

Figure 4: Estimated utility functions and preference cdfs for four sectors, with 90% bootstrap intervals and bootstrap replications.

Elasticities under Nonlinear Pricing

The identified and estimated utility and heterogeneity functions permit computation of elasticities under arbitrary nonlinear price counterfactuals. The authors consider both uniform level (“price inflation”) and curvature (“rebate/discount structure”) perturbations:

• For a parametric perturbation $u(\cdot)$9, the elasticity is expressed as

$\varepsilon$0

• Under uniform-level ($\varepsilon$1) and power transformation ($\varepsilon$2), the elasticities are evaluated using the estimated functional objects.

Empirical results reveal a pattern: low-volume buyers show higher price-sensitivity, while high-volume buyers are inelastic, consistent with limited outside options at scale.

Figure 5: Estimated baseline level elasticity ($\varepsilon$3) for four sectors with 90% bootstrap confidence bands.

Figure 6: Estimated curvature elasticity ($\varepsilon$4) for four sectors with 90% bootstrap confidence bands.

Extensions: Endogenous Prices and Covariates

The methodology is extended to accommodate potentially endogenous prices, through semiparametric IV estimation of the price schedule conditional on covariates $\varepsilon$5. An instrumental variables identification hinges on exogenous heterogeneity affecting the utility but not pricing. The framework also generalizes for discrete observed covariates (stratified estimation) and in principle can be extended to high-dimensional continuous covariates, though the asymptotic theory for the latter is not developed here.

Figure 7: Covariate-conditional estimation of $\varepsilon$6 and $\varepsilon$7 for three strata of $\varepsilon$8 in simulated data ($\varepsilon$9), showing adaptation to observed heterogeneity.

Discussion and Implications

The approach breaks the identification impasse for nonseparable consumer preference models under nonlinear budget constraints, using the discrete variation induced by policy, regulation, or market design to separate utility and heterogeneity nonparametrically. The regularized iterative identification/estimation strategy is novel in the econometric literature of inverse problems. The developed estimator achieves near-parametric rates and bootstrap inference, suitable for empirical deployment with large administrative datasets.

Critical implications include:

• Possibility for fully nonparametric welfare and elasticity analysis in nonlinear pricing environments (utilities, telecom, insurance, taxation).
• Relaxation of functional form assumptions endemic to classic structural estimation under piecewise-linear or nonlinear tariffs.
• Direct computation of counterfactual elasticities under arbitrary price schedule changes, supporting robust policy analysis.

Theoretical limitations: Identification hinges on stable preference distributions across regimes, exogeneity of price schedule variation, and sufficient overlap of the support. Regularization parameter selection (the number of iterations $\max_Q u(Q) + Q\varepsilon - \tau(P(Q) - B)$0) is currently heuristic; automated selection remains open.

Practical limitations: The method requires repeated cross-sections under policy variation. In empirical exercises with only a single regime or shifting heterogeneous distributions, identification fails.

Conclusion

The authors’ framework endows empirical researchers with a constructive, nonparametric toolkit for the identification and estimation of structural consumer preferences in the presence of nonlinear pricing, leveraging repeated observational data and minimal functional restrictions. The iterative scheme and associated bootstrap procedures allow for credible finite-sample inference and counterfactual analysis, bridging substantial gaps between theoretical nonparametric identification and empirical practice in demand estimation. The main theoretical apparatus and empirical findings motivate several avenues for future research, including automated regularization selection, further generalization to multiple regimes, and extension to high-dimensional covariate settings.