arxiv: 2604.25507 · v1 · submitted 2026-04-28 · 💰 econ.EM

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

Samuele Centorrino , Fr\'ed\'erique F\`eve , Jean-Pierre Florens

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Pith reviewed 2026-05-07 14:05 UTC · model grok-4.3

classification 💰 econ.EM

keywords nonparametric identificationnonlinear pricingconsumer preferencesunobserved heterogeneityquantile functionfunctional equationiterative estimationeconometrics

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The pith

With sufficient variation across price schedules, the utility function and distribution of consumer preference types can be nonparametrically identified.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a nonparametric method for recovering consumer preferences from choices under nonlinear pricing using repeated observations. It demonstrates that both the utility function and the distribution of unobserved preference types are identifiable by treating the quantile function of types as the solution to a functional equation. This is useful because nonlinear pricing is common in many industries, and nonparametric identification avoids restrictive assumptions that could bias policy recommendations. The method includes an iterative estimation procedure with favorable convergence properties and extends to endogenous prices.

Core claim

What carries the argument

The quantile function of unobserved preference types, defined as the solution to a functional equation derived from observed demand under varying nonlinear price schedules.

If this is right

The utility function is recovered nonparametrically for all relevant income levels.
The distribution of unobserved preference heterogeneity is fully identified.
Estimation achieves near-parametric rates through iteration.
Bootstrap procedures provide valid inference.
Extensions handle price endogeneity and observed covariates.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This approach could enable nonparametric analysis of welfare effects from changes in nonlinear tariffs.
It might be adapted to other repeated choice settings with varying menus, like insurance or subscription services.
Sufficient variation in price schedules could be verified empirically by subsample stability.
Future work could incorporate dynamics if choices are observed over time.

Load-bearing premise

There is sufficient independent variation in the multiple price schedules observed by consumers to trace out the utility function and type distribution.

What would settle it

If estimates of the utility function obtained from different combinations of price schedules are inconsistent or fail to satisfy monotonicity conditions in an empirical application, the identification would not hold.

Figures

Figures reproduced from arXiv: 2604.25507 by Fr\'ed\'erique F\`eve, Jean-Pierre Florens, Samuele Centorrino.

**Figure 2.** Figure 2: Estimation of the quantile function of 𝜀 (a) and the utility function (b) for Design 1 for 20 randomly selected sample of size 𝑛 = 1000. α Λ( α) 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 ΛˆN Λ (a) Estimation of Λ𝜀 Q u(Q) 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 uˆN u (b) Estimation of 𝑢 view at source ↗

**Figure 3.** Figure 3: Estimation of the quantile function of 𝜀 (a) and the utility function (b) for Design 2 for 20 randomly selected sample of size 𝑛 = 1000. 25 view at source ↗

**Figure 4.** Figure 4: Estimation of the quantile function of 𝜀 (a) and the utility function (b) for Design 1 for a randomly selected sample of size 𝑛 = 1000. α Λ( α) 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 ΛˆN Λ 95% bootstrap CI 95% MC CI (a) Estimation of Λ𝜀 Q u(Q) 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 uˆN u 95% bootstrap CI 95% MC CI (b) Estimation of 𝑢 view at source ↗

**Figure 5.** Figure 5: Estimation of the quantile function of 𝜀 (a) and the utility function (b) for Design 2 for a randomly selected sample of size 𝑛 = 1000. 26 view at source ↗

**Figure 6.** Figure 6: Estimation of 𝑢 (left) and 𝐹𝜀|𝑋 (right) for the four selected sectors. The dashed lines are the 90% bootstrap confidence intervals. The grey lines are 20 random bootstrap realizations. 9.1 Demand Elasticities Once the utility function 𝑢(𝑄; 𝑋) and the price schedules 𝑃 (𝑄) have been estimated, one of the potential objects of interest are demand elasticities. Under nonlinear pricing, the notion of a price el… view at source ↗

**Figure 7.** Figure 7: Baseline level elasticity ( view at source ↗

**Figure 8.** Figure 8: Baseline curvature elasticity (𝜎 = 1) for four sectors with 90% bootstrap confidence bands (dashed). Bands are omitted when bootstrap results are not yet available. ing restrictions whose difference yields a linear iterative functional equation in the quantile function of unobserved preference heterogeneity. We show that this equation has a unique solution under mild regularity conditions, derive a closed-… view at source ↗

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives a nonparametric identification route for utility and type distributions under nonlinear pricing by turning the quantile function into a functional equation solved from multiple schedules, plus an iterative estimator with near-parametric rates.

read the letter

The core contribution is a way to recover both the utility function and the distribution of unobserved preference types nonparametrically when the same consumers are observed under several different nonlinear price schedules. The quantile function of types is expressed as the solution to a functional equation, and the authors lay out conditions under which this pins down the objects uniquely. They then build an iterative estimator whose bias shrinks exponentially while variance grows only polynomially, which is a nice technical feature compared with standard nonparametric rates. Monte Carlo results and an application to European mail data illustrate that the procedure recovers the targets in finite samples. They also add a bootstrap and extensions for price endogeneity and observed covariates. That package is what makes the work worth looking at for applied demand work in tariff settings. The main soft spot is the identification argument itself. It requires sufficient independent variation across the observed price schedules that is not correlated with the unobserved types. If the schedules are chosen by the firm with knowledge of consumer heterogeneity, or if their support is too narrow relative to the dimension of types, the functional equation can fail to deliver uniqueness. The paper states that it derives the necessary conditions, but those conditions are the part that will need the most scrutiny in any application. Without seeing the full proofs it is hard to judge how restrictive they turn out to be in practice. This is aimed at researchers estimating demand under quantity discounts or two-part tariffs in regulated sectors such as utilities, telecom, or postal services. Anyone who wants to relax parametric assumptions on heterogeneity while still getting usable estimates will find the functional-equation approach and the rate result useful. It is coherent enough and addresses a genuine identification gap, so it deserves a serious referee. I would send it out, but the referees should be asked to check whether the maintained conditions on schedule variation are plausible in the empirical example and how sensitive the recovered utility and distributions are to small changes in those conditions.

Referee Report

2 major / 3 minor

Summary. The paper develops a nonparametric identification strategy for consumer utility functions and the distribution of unobserved preference types from repeated choice data under nonlinear pricing. It shows that the quantile function of unobserved types solves a functional equation derived from optimal bundle choices across multiple price schedules, derives conditions for unique identification, proposes an iterative estimator achieving near-parametric convergence rates via exponentially decaying regularization bias, provides a bootstrap for inference, and extends the framework to price endogeneity and observed heterogeneity. The approach is illustrated via Monte Carlo simulations and an empirical application to European mail carrier data.

Significance. If the identification conditions hold, the paper makes a substantial contribution to the econometric literature on demand estimation under nonlinear tariffs by delivering nonparametric recovery of both preferences and heterogeneity without strong functional form assumptions. The near-parametric rate of the iterative estimator and the valid bootstrap are technically attractive features, as is the explicit handling of endogeneity in an extension. The Monte Carlo and empirical results provide concrete evidence of finite-sample performance in a relevant setting.

major comments (2)

[Identification section (functional equation and conditions)] The nonparametric identification result hinges on the functional equation for the quantile function of types admitting a unique solution, which in turn requires sufficient independent variation across observed price schedules that is uncorrelated with unobserved preference heterogeneity. The manuscript states that conditions ensuring identification are derived, but the precise statement of these conditions (including invertibility of the type-to-bundle mapping and the support requirements on the schedule distribution) is central to the claim and would benefit from a more explicit theorem statement with verifiable primitives.
[Estimation and rate analysis] The iterative estimator is presented as delivering near-parametric rates because regularization bias decays exponentially while variance grows only polynomially. However, the manuscript should clarify the precise dependence of the rate on the number of iterations, the choice of regularization parameter, and the dimension of heterogeneity, as these details are load-bearing for the rate claim and its comparison to standard nonparametric estimators.

minor comments (3)

[Abstract] The abstract claims 'near-parametric convergence rate' without stating the exact rate; adding the rate (e.g., n^{-1/2} up to logs) would improve clarity for readers.
[Monte Carlo simulations] In the Monte Carlo section, the design of the price schedules and the data-generating process for types should be described in sufficient detail to allow readers to verify that the simulated variation satisfies the identification conditions.
[Inference section] The bootstrap procedure is stated to be valid, but a brief discussion of the resampling scheme (e.g., whether it resamples schedules or consumers) would help assess its robustness to the dependence structure induced by repeated observations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of our paper and the constructive suggestions. We agree that greater explicitness in the identification theorem and the rate analysis will improve readability. We address each major comment below and will revise the manuscript to incorporate the clarifications.

read point-by-point responses

Referee: The nonparametric identification result hinges on the functional equation for the quantile function of types admitting a unique solution, which in turn requires sufficient independent variation across observed price schedules that is uncorrelated with unobserved preference heterogeneity. The manuscript states that conditions ensuring identification are derived, but the precise statement of these conditions (including invertibility of the type-to-bundle mapping and the support requirements on the schedule distribution) is central to the claim and would benefit from a more explicit theorem statement with verifiable primitives.

Authors: We agree that consolidating the identification conditions into a single, self-contained theorem statement will make the result easier to verify. The manuscript already derives the required primitives (invertibility of the type-to-bundle mapping under the maintained assumptions on utility and the support conditions on the distribution of price schedules that ensure sufficient independent variation uncorrelated with unobserved heterogeneity). In the revision we will add an explicit theorem that lists these primitives verbatim and states the unique solution property of the functional equation for the quantile function. revision: yes
Referee: The iterative estimator is presented as delivering near-parametric rates because regularization bias decays exponentially while variance grows only polynomially. However, the manuscript should clarify the precise dependence of the rate on the number of iterations, the choice of regularization parameter, and the dimension of heterogeneity, as these details are load-bearing for the rate claim and its comparison to standard nonparametric estimators.

Authors: We thank the referee for highlighting the need for greater precision on the rate. The paper establishes that the regularization bias decays exponentially in the number of iterations while the variance term grows only polynomially in the sample size, yielding a near-parametric rate. In the revision we will add an explicit corollary that states the convergence rate as a function of the iteration count T, the regularization parameter, and the dimension of heterogeneity, showing how T is chosen to make the bias term negligible relative to the parametric rate and why the exponential decay ensures the overall rate remains near-parametric regardless of dimension under the maintained conditions. revision: yes

Circularity Check

0 steps flagged

No circularity: identification via functional equation from observed price schedule variation is self-contained

full rationale

The paper's core claim is that the quantile function of unobserved types solves a functional equation derived from consumer choices under multiple observed nonlinear price schedules, with identification ensured by sufficient independent variation across schedules. This setup draws directly from the data-generating process and standard consumer theory without reducing to self-definition, fitted parameters renamed as predictions, or load-bearing self-citations. The iterative estimator is described separately with its own convergence properties. No quoted steps in the provided text exhibit any of the enumerated circular patterns; the derivation remains independent of its target objects.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard consumer theory assumptions of utility maximization under budget constraints and sufficient exogenous variation in price schedules across observations; no new free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Consumers maximize a utility function subject to nonlinear budget constraints.
Implicit foundation for deriving the functional equation from choice behavior.
domain assumption There exists sufficient independent variation in price schedules across repeated observations.
Required for tracing out the utility function and type distribution nonparametrically.

pith-pipeline@v0.9.0 · 5439 in / 1328 out tokens · 66667 ms · 2026-05-07T14:05:49.008886+00:00 · methodology

discussion (0)

Reference graph

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