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Hedonic Habits: The Empirical Content of Dynamic Hedonic Models
\affil[ ]{MIT}
Many product attributes---such as sugar, nicotine, ethanol, and caffeine---are habit forming. Yet empirical hedonic valuation, following gorman_possible_1956 and lancaster_new_1966, almost universally treats preferences as time separable.\footnote{Hedonic models are central in empirical industrial organisation and applied micro; see, e.g., smith_value_1986, heckman_importance_1987, berry_automobile_1995, nevo_measuring_2001, gibbons_valuing_2003, bajari_estimating_2005, and greenstone_does_2008.} When some characteristics are habit forming, time separability is no longer an innocuous normalisation but a restriction on the economic meaning of hedonic prices and choices. This raises two questions: when do observed prices and choices admit a coherent characteristics-based interpretation at all, and when does allowing for habits change the admissible interpretation of hedonic prices?
Whether hedonic willingness-to-pay (WTP) retains its usual interpretation in the presence of habit formation matters because it is routinely used for welfare analysis and policy evaluation. In settings such as sugar taxation, alcohol regulation, tobacco policy, and environmental product standards, researchers recover marginal valuations for attributes under time-separable preferences and use them to rank counterfactuals.\footnote{For examples valuing potentially habit-forming food attributes under static hedonic preferences, see dubois_how_2020, haeck_estimating_2022, and le_fur_willingness_2022.} If some attributes are habit forming, however, observed prices may reflect both contemporaneous utility and the continuation value generated by current consumption. In that case, static hedonic WTP need not measure the object it is interpreted as measuring.
Existing models of habit formation are typically formulated over goods, not characteristics. A large literature studies intertemporal dependence in the consumption of cigarettes, alcohol, and digital products by modelling utility as a function of past and current quantities becker_theory_1988,gruber_tax_2004,demuynck_ill_2013,crawford_habits_2010,allcott_gentzkow_song_2022. These models characterise dynamic dependence in product quantities, but they do not determine whether prices admit a coherent interpretation as valuations of underlying attributes. When valuation and policy analysis are conducted in characteristics space, the dynamic structure must be specified at the level of attributes rather than goods. This is particularly important when reinforcement operates at the level of sugar, caffeine, or nicotine rather than at the level of the composite product itself. Understanding when a dynamic hedonic interpretation is coherent is therefore necessary to discipline applied valuation.
This paper provides a nonparametric framework for evaluating when dynamic hedonic interpretation is economically coherent. I make three contributions. First, I derive necessary and sufficient revealed-preference (RP) conditions---in the spirit of samuelson_consumption_1948, houthakker_revealed_1950, afriat_construction_1967, and browning_nonparametric_1989---under which observed prices and choices admit a coherent dynamic characteristics-based interpretation when some attributes are habit forming. Second, I decompose empirical failure into two conceptually distinct margins: (i) structural feasibility of a hedonic shadow-price representation given the maintained goods-to-characteristics technology, and (ii) behavioural consistency of intertemporal choice conditional on that technology. Third, I develop computationally tractable, distance-based diagnostics that quantify how close the data are to satisfying each margin, thereby disciplining when static hedonic valuation is economically defensible and when a dynamic reinterpretation becomes admissible.
The key insight is that dynamic hedonic rationalisability has a two-stage structure. First, observed prices must admit a low-dimensional shadow-price representation implied by the goods-to-characteristics mapping; this is a structural restriction on the admissible interpretation of observed prices, independent of intertemporal optimisation. Second, conditional on such a representation existing, the implied characteristic shadow prices must rationalise observed choices over time; this is a behavioural restriction. Separating these margins clarifies whether a model fails because the maintained characteristics technology cannot sustain a hedonic interpretation or because intertemporal optimisation is violated conditional on that technology.
To formalise these ideas, I study the RP implications of a dynamic hedonic environment in which a single consumer purchases goods over time at some observed prices. The RP exercise is conducted consumer by consumer, so the framework permits unrestricted heterogeneity across households in valuations over characteristics, discounting, and habit formation. These observed goods map into characteristics through a maintained technology, and utility depends on both contemporaneous and lagged levels of selected characteristics. Some attributes are habit forming, so current consumption affects future marginal utility. The analysis characterises the observable restrictions on prices and choices required for a coherent dynamic characteristics-based interpretation.
A useful way to organise the framework is as a two-by-two comparison between goods and characteristics, on the one hand, and static and dynamic preferences, on the other. To fix ideas, consider the setting of a single household making repeated cereal purchasing decisions that I return to in the empirical application. In this setting, a static goods-based model treats each barcode-level box as a separate object and asks only whether observed purchase paths are rationalisable over goods. A dynamic goods-based model can rationalise persistence in repeated purchases, but it attaches that persistence to cereal products themselves rather than to the underlying attributes they bundle. A static characteristics-based model instead imposes the additional discipline that observed goods prices must be representable through shadow values on measurable attributes such as sugar, sodium, and taste-related characteristics, but rules out the possibility that current consumption of those attributes shifts future marginal utility. The dynamic characteristics-based model developed here combines both forms of discipline.
This two-by-two comparison helps clarify both what the dynamic characteristics model can accommodate and what it still rules out. On the behavioural side, allowing for habits can rationalise intertemporal patterns that a static characteristics model cannot. For example, a household that consumes a high-sugar cereal today may switch tomorrow toward a lower-sugar bundle even if the high-sugar product becomes cheaper, because current exposure can reduce the future marginal value of sugar through fatigue or satiation. On the structural side, however, dynamics do not relax the price restrictions implied by characteristics space. If two cereals with very similar measured attributes have observed prices that cannot be represented by common shadow values on those attributes, a goods-based model may still rationalise the purchase data by treating them as distinct products, whereas a characteristics-based model cannot. In this sense, habits expand the class of admissible behavioural patterns without relaxing the structural discipline that distinguishes characteristics-based valuation from goods-based demand.
My main theoretical result is a dynamic Afriat-type RP characterisation that extends static characteristics-based rationalisability (e.g., blow_revealed_2008) to accommodate habit formation. The test asks whether there exist latent characteristic shadow prices and a discount factor such that two things hold: first, observed goods prices admit the shadow-price representation implied by the maintained goods-to-characteristics technology and, second, observed choice sequences satisfy the corresponding dynamic RP inequalities, ruling out cycles in revealed marginal valuations over time. This distinction yields two economically different failure modes: structural failure, when no admissible hedonic shadow-price system exists, and behavioural failure, when such a system exists but observed intertemporal choices are not dynamically rationalisable. I then complement the binary test with distance-based diagnostics that quantify how far the data lie from each margin of consistency.
This decomposition into structural and behavioural margins delivers a new taxonomy of empirical failure and quantitative measures of its severity. Rather than asking only whether the model passes, the framework asks how far the data lie from consistency on each margin and in what sense. On the structural side, the relevant object is the hedonic price manifold implied by the maintained goods-to-characteristics technology: the distance statistic measures the minimum joint adjustment to observed prices, in Euclidean price space, required to place the purchased-good price vector on that manifold. On the behavioural side, the relevant object is the set of dynamic RP inequalities conditional on the implied shadow prices: the diagnostic is a Critical Cost Efficiency Index, which measures the smallest proportional relaxation of those inequalities required for observed choices to become rationalisable, or equivalently, the amount of slack needed to eliminate dynamic cycles in revealed marginal valuations. These statistics therefore distinguish small departures from consistency from large ones and show whether empirical failure comes primarily from the price representation or from intertemporal choice.
I apply the framework to household-level scanner data on cold-cereal purchases. In general, my framework accommodates any setting in which repeated purchases, observed expenditures, and measured product characteristics make it possible to study dynamic hedonic rationalisability at the relevant subject level. The empirical implementation uses only purchased-good prices and therefore relies on a missing-price version of my test. I find that most of the large differences in raw pass rates between goods-based and characteristics-based models reflect the structural restrictions imposed by the hedonic price representation. At the same time, the associated distance-to-manifold statistics show that these structural violations are often modest in magnitude. Conditional on satisfying those structural restrictions, the behavioural diagnostics also show that allowing for habit formation improves intertemporal coherence relative to static characteristics models for a subset of households.
My results clarify when dynamics matter for hedonic valuation. If observed prices do not admit a characteristics-based interpretation under the maintained technology, a hedonic valuation exercise is not economically coherent. If the structural restrictions hold but intertemporal separability fails, a dynamic hedonic interpretation is admissible and static valuation may be misinterpreted when dynamics are ignored. The framework therefore provides a disciplined diagnostic for applied work: it distinguishes structural misspecification from behavioural misspecification, clarifies when static hedonic valuation is economically defensible, and identifies where additional structure would be needed to recover unique welfare objects.
The remainder of the paper is organised as follows. After situating the paper within the related literature, Section 2 introduces the dynamic hedonic model and derives the necessary and sufficient RP characterisation. All proofs are in Appendix (ref). Section 3 presents corollaries showing that static hedonic and dynamic goods-based models arise as special cases of my framework. Section 4 applies the theory to household scanner data on cereal purchases.
\noindentRelated work
This paper is most closely related to two nonparametric RP literatures that have developed separately: tests of preferences over characteristics blow_revealed_2008 and RP analyses of intertemporal dependence crawford_habits_2010, both rooted in the foundational work of afriat_construction_1967, diewert_afriat_1973, and varian_nonparametric_1982. Relative to the former, I introduce habits into characteristics space; relative to the latter, I impose dimensionality reduction through a maintained goods-to-characteristics technology. Static hedonic RP tests and dynamic goods-based RP tests therefore arise as special cases of my framework. The main contribution is to characterise how these two sources of discipline interact, yielding a new taxonomy of empirical failure and tractable diagnostics for its severity.
The paper also relates to the hedonic valuation literature following rosen_hedonic_1974.\footnote{rosen_hedonic_1974's (rosen_hedonic_1974) hedonic method has been widely applied across housing, environmental economics, health, education, and industrial organisation; see, for example, smith_value_1986, gibbons_valuing_2003, bajari_estimating_2005, and greenstone_does_2008.} My point of departure from that literature is the maintained assumption of time-separable preferences over characteristics. When some attributes are habit forming, observed prices need not map cleanly into contemporaneous marginal valuations, so the economic interpretation of hedonic WTP becomes an empirical question rather than a maintained one. In that sense, the paper complements structural hedonic approaches such as bajari2005demand: rather than recovering preference parameters from equilibrium price schedules, I ask when a dynamic hedonic interpretation of observed prices is itself empirically coherent.
The paper is also related to the rational addiction literature initiated by becker_theory_1988.\footnote{Empirical applications have largely focused on goods such as alcohol, cigarettes, caffeine, and illicit drugs; see, for example, becker_empirical_1994, grossman_empirical_1998, gruber_tax_2004, and demuynck_ill_2013.} The central difference is that the canonical rational-addiction framework is formulated at the level of goods and is primarily theoretical, whereas my framework allows habits to operate at the level of characteristics and delivers an empirically implementable RP characterisation. This distinction matters when persistence is more naturally attached to attributes such as sugar, sodium, nicotine, or caffeine than to the composite products that bundle them koob_drug_1997. Relative to empirical work on addiction over nutrient profiles richards_native_2006, my approach replaces functional-form restrictions with a nonparametric RP characterisation.
Finally, the framework is distinct from models of inventory behaviour and stockpiling in consumer demand hendel_sales_2006. Storage can generate persistence in purchases under time-separable preferences, whereas habit formation in my setting operates through state dependence in the flow utility function over characteristics. The empirical framework therefore treats persistence induced by habits as conceptually different from persistence induced by intertemporal substitution, even though distinguishing the two in purchase data requires a maintained link between purchases and the underlying consumption state.
Taken together, the paper shows how intertemporal dependence, dimensionality reduction, and RP discipline jointly determine whether hedonic valuation is economically meaningful. Moving to characteristics space imposes demanding structural restrictions on the admissible price system; allowing for dynamics then changes the set of admissible behavioural patterns conditional on satisfying those restrictions.
I begin by formalising a dynamic hedonic environment in which goods map into characteristics and habits attach to a subset of those characteristics. I observe a single consumer for $t=1,\ldots,T$, with purchases $\bm{x}_t\in\mathbb{R}^K_+$ and present-value prices $\bm{\rho}_t\in\mathbb{R}^K_+$. Observed goods prices are taken as given from the consumer's perspective, so the object of the analysis is not price determination but whether observed choices can be rationalised. Goods map into $J$ measured characteristics via a time-invariant linear technology $\bm{z}_t=\bm{A}\bm{x}_t$ following gorman_possible_1956, where $\bm{A}$ is a $J\times K$ matrix (typically $J<K$).\footnote{I adopt a linear transformation from goods to characteristics space as it is the most widely used specification. Most results extend to a non-linear setting where $\bm{z}=\bm{F}(\bm{x})$ is increasing and strictly concave; see Appendix (ref) for an analogue of the consistency definition. The key difference to the linear case arises in the marginal product: whereas $\partial \bm{z}/\partial \bm{x}=\bm{A}^{\prime}$ is constant in the linear model, the marginal product varies with demand under a non-linear transformation.} I treat $\bm{A}$ as known and stable over time; it captures objectively defined product attributes, while valuation is encoded in preferences. Although the model is written for one consumer, the empirical RP exercise is applied household by household, so valuations over characteristics, discounting, and habit formation are all allowed to vary freely across consumers.
I partition characteristics as $\bm{z}_t=((\bm{z}_t^c)^{\prime},(\bm{z}_t^a)^{\prime})^{\prime}$, where $\bm{z}_t^c\in\mathbb{R}^{J_1}$ are non-habit-forming characteristics and $\bm{z}_t^a\in\mathbb{R}^{J_2}$ are habit-forming characteristics, with $J_1+J_2=J$. The analyst defines this partition. Crucially, this formulation allows intertemporal dependence to operate at the level of attributes rather than goods, so persistence in behaviour need not be attributed to non-habit-forming components bundled within a product. Although preferences are defined over characteristics, choice and budget constraints remain in goods space.
To clarify the economic content of the model, it is useful to fix ideas with a simple example. Suppose the consumer chooses between two cereal products, where each good bundles two measurable attributes: a contemporaneous “taste” characteristic (e.g., nutty-ness) and a habit-forming “sensory” characteristic (e.g., salt or sugar intensity). The key modelling choice is that habits attach to the latter attribute rather than to the cereal good itself: consuming a high-intensity product today can change tomorrow's marginal value of intensity (due to sensory fatigue or craving), even if the consumer switches cereal product. In this example, the model's structural content is that observed goods prices must be representable as shadow values on attributes given $\bm{z}_t=\bm{A}\bm{x}_t$, while its behavioural content is that those shadow values must admit a concave, dynamically consistent utility representation.
Preferences are represented by a felicity (i.e., flow utility) function $u:\mathbb{R}^{J+J_2}\to\mathbb{R}$ that depends on current characteristics and one lag of the habit-forming subset, $u(\bm{z}_t^c,\bm{z}_t^a,\bm{z}_{t-1}^a)$, as in the one-lag “short memory habits” specification of boyer_habit_1978,boyer_rational_1983 and becker_empirical_1994. The multi-lag extension is straightforward and deferred to \href{https://joauer-mit.github.io/when-do-habits-matter-appendix/online_appendix_dynamics_hedonic_val.pdf}{Online Appendix E}. I assume quasi-linearity in an outside good $y_t$ with unit price, as is standard in empirical IO and hedonic demand models for narrow product categories berry_automobile_1995,nevo_measuring_2001. I also maintain local non-satiation, concavity, and superdifferentiability of $u$; I impose no further sign or monotonicity restrictions on the habit-forming components. I refer to this environment as the habits-over-characteristics model.
The consumer chooses $\{(\bm{x}_t,y_t)\}_{t=1}^T$ to solve
where $\beta\in(0,1]$ is a discount factor and $W$ is present-value lifetime wealth. I use the augmented notation
so $\bm{\tilde{z}}_t\in\mathbb{R}^{J+J_2}$, $\bm{\tilde{x}}_t\in\mathbb{R}^{2K}$, and $\bm{\tilde{A}}$ is a $(J+J_2)\times 2K$ block matrix. Throughout, I treat $\bm{x}_0$ (equivalently, the initial habit stock $\bm{z}_0^a$) as fixed and exogenous. Because the model has one lag and a finite horizon, there is no continuation term beyond $T$; equivalently, I set $\bm{\pi}_{T+1}^1 \equiv \bm{0}$ by convention. The key question is whether the observables $\{(\bm{\rho}_t,\bm{x}_t)\}_{t=1}^T$ can be rationalised by (ref), and, if so, what testable restrictions this imposes on prices and choices.
I now ask whether the observed data can be rationalised by optimising behaviour under the habits-over-characteristics model. This notion of consistency encompasses both the existence of a hedonic shadow-price representation (a structural requirement) and the coherence of intertemporal behaviour conditional on that representation. Here “structural” does not refer to price determination. Rather, it refers to whether the maintained goods-to-characteristics technology can rationalise observed prices through a corresponding system of characteristic shadow prices.
The following lemma provides necessary and sufficient conditions for consistency.
Proof: See Appendix (ref). $\qed$
Lemma (ref) links observed discounted market prices to shadow prices that measure the consumer's discounted marginal valuations of characteristics gorman_possible_1956. Current goods prices therefore reflect both contemporaneous utility from characteristics and the intertemporal effects induced by habit formation. In the running cereal example above, the price of a high-salt or high-sugar cereal must reflect not only the consumer's current taste for nutty-ness and sensory intensity, but also how today's intensity alters tomorrow's marginal utility of that same sensory characteristic. The first-order condition $(\star)$ formalises this intuition: goods prices equal the sum of contemporaneous shadow values and the discounted continuation value generated by habit-forming attributes.
Formally, the shadow price $\bm{\pi}_t^0$ can be interpreted as the discounted marginal valuation of contemporaneous characteristics, while $\bm{\pi}_t^{1}$ captures the marginal utility impact of past consumption of habit-forming characteristics. The key economic implication is a price wedge: current goods prices internalise future utility effects whenever habits are present. When lagged consumption lowers future marginal utility (i.e., $\partial_{\bm{z}_t^a} u(\bm{\tilde{z}}_{t+1}) < 0$), goods prices satisfy \[ \rho_t^k = \bm{a}_k^{\prime}\bm{\pi}_t^0 + \bm{a}_k^{a\prime}\bm{\pi}_{t+1}^{1} < \bm{a}_k^{\prime}\bm{\pi}_t^0, \] so ignoring intertemporal dependence understates contemporaneous WTP for the current characteristics bundled in goods with negatively reinforcing attributes, such as sensory fatigue. If lagged consumption instead raises future marginal utility, the inequality reverses. This decomposition is generically set-identified, as discussed below, so the wedge should be interpreted as a theoretical mapping from prices to admissible marginal valuations rather than as a point-identified empirical object absent further structure.
A further implication is that under a linear characteristics technology, observed prices for goods consumed in strictly positive amounts must lie in the column space of the technology matrix. By complementary slackness, the first-order conditions bind on the support of consumption, so the intertemporal budget constraint holds equivalently when expressed in goods space or in characteristics space with shadow prices: \[ \sum_{t=1}^{T}\bm{\rho}_t^{\prime}\bm{x}_t = \sum_{t=1}^{T}\left( \bm{z}_t^{\prime}\bm{\pi}_t^0 + \bm{z}_t^{a\prime}\bm{\pi}_{t+1}^{1} \right), \] with $\bm{\pi}_{T+1}^1\equiv\bm{0}$ by the terminal convention above.
In sum, the model’s empirical content is governed by two restrictions: a structural requirement that observed goods prices admit characteristic shadow prices consistent with the maintained goods-to-characteristics technology, and a behavioural requirement that these shadow prices be consistent with concave, dynamically stable preferences.
I now state the central theoretical result of the paper. It shows that the model has two distinct sources of empirical content: a structural requirement that observed goods prices admit characteristic shadow prices given $\bm{A}$, and a behavioural requirement that these shadow prices be consistent with concave, dynamically stable preferences.
Proof: See Appendix (ref). $\qed$
Theorem (ref) delivers a complete RP characterisation of the one-lag habits-over-characteristics model: the data are rationalisable if and only if there exist shadow prices and a discount factor satisfying conditions (ref)--(ref). When such objects exist, one can construct a concave, locally non-satiated utility function over characteristics that rationalises observed choices; when they do not, no such representation is possible. Given the boundary convention stated above, the final-period pricing restriction is simply (ref)--(ref) with $\bm{\pi}_{T+1}^1=\bm{0}$. The theorem therefore delivers a sharp, nonparametric test of dynamic consistency in characteristics space.
It is useful to interpret the economic content of the three conditions. Condition (ref) imposes cyclical monotonicity on the shadow prices. Economically, it rules out “cycles” in revealed marginal valuations over augmented characteristic bundles: there should be no sequence of observed trades in characteristics space that would allow a costless improvement by returning to the starting point. Formally, cyclical monotonicity is equivalent to concavity of the instantaneous utility function rockafellar_convex_1970. This condition is precisely the behavioural discipline of the model.
Conditions (ref) and (ref) impose the structural pricing restrictions implied by the habits-over-characteristics model. They require that observed goods prices be representable as linear combinations of contemporaneous and forward-looking shadow prices. Economically, current goods prices must internalise both current marginal utility from characteristics and the continuation value induced by habit formation. In the running cereal example, the price of a high salt or sugar cereal must reflect not only current taste for nutty-ness and sensory intensity, but also how today's sensory intensity alters tomorrow's marginal utility. These equalities therefore encode the intertemporal wedge introduced by past consumption directly into the price system.
Finally, note that the mechanism tested here differs conceptually from inventory-driven persistence hendel_sales_2006: stockpiling generates serial correlation in purchases through intertemporal substitution and storage under time-separable preferences. Contrastingly, persistence in my framework operates through state dependence in utility over characteristics, so current consumption of habit-forming attributes carries a continuation value by shifting future marginal valuations. In purchase data, however, the two mechanisms need not be empirically separable without additional structure linking purchases to consumption.
Testing consistency reduces to an empirical search for shadow prices and a discount factor satisfying (ref)--(ref). The system is nonlinear jointly in shadow prices and $\beta$, but becomes linear conditional on $\beta$. For any fixed discount factor, feasibility can therefore be assessed via a linear programme. Repeating this feasibility check over a grid of candidate discount factors yields a computationally straightforward implementation strategy.
A practical complication arises from condition (ref). In its raw form, cyclical monotonicity requires the inequalities to hold for all finite ordered cycles of observations, which quickly becomes computationally burdensome as $T$ grows. \href{https://joauer-mit.github.io/when-do-habits-matter-appendix/online_appendix_dynamics_hedonic_val.pdf}{Online Appendix G} derives an equivalent linear-programming formulation based on Afriat inequalities, replacing this cycle condition with a quadratic number of pairwise constraints in $T$.
Interpreting the Afriat test requires care. A positive result establishes existence: there exists some concave utility function and discount factor consistent with the data. The representation, however, is not unique. Distinct utility functions---beyond simple monotone transformations---may rationalise the same dataset. Moreover, rationalisability depends on the level of temporal and product aggregation; for example, time aggregation may smooth consumption in a way that mimics habit persistence.
A negative result is likewise not diagnostic of the precise source of failure. Rejection may reflect habit persistence extending beyond one lag, non-concavities in preferences, misspecification of the characteristics technology, or an incorrect partition of $\bm{A}$ into contemporaneous and habit-forming components (for instance, treating “taste” preferences like nutty-ness as static when the data in fact suggest that exposure today shifts future marginal valuations). The Afriat test should therefore be interpreted as a sharp but reduced-form diagnostic of dynamic rationalisability, rather than as a structural identification device.
Although Theorem (ref) provides a complete characterisation, its direct implementation can become computationally burdensome as the dimension of goods, characteristics, or time increases. This subsection therefore derives a low-dimensional diagnostic implied by consistency: a necessary rank condition that can be checked period by period. Failure of the condition immediately falsifies the model, while satisfaction is necessary but not sufficient.
The restriction is structural in the sense that it concerns only whether the maintained goods-to-characteristics technology can in principle support any shadow-price representation of observed prices. If the condition fails, no candidate preferences---static or dynamic---can rationalise the data because the implied shadow-price system does not exist.
Let $\bm{\rho}_t^+$ denote the $K_t^+\le K$-dimensional vector of discounted prices of goods consumed in strictly positive quantities in period $t$. Let $\bm{B}_t$ denote the $J\times K_t^+$ submatrix of $\bm{A}$ collecting the contemporaneous characteristics of those goods, and let $\bm{B}_t^a$ denote the $J_2\times K_t^+$ submatrix collecting the corresponding habit-forming characteristics. For interior dates, Theorem (ref) implies that
where $\bm{\tilde{B}}_t$ is the $K_t^+\times (J+J_2)$ augmented technology matrix. Because $\bm{B}_t^a$ is formed by rows of $\bm{B}_t$, the augmented matrix $\bm{\tilde{B}}_t$ has the same column space as $\bm{B}'_t$, so habit formation tilts shadow prices within the same $J$-dimensional price manifold rather than expanding it. At a genuine terminal date, the same pricing equation holds with $\bm{\pi}_{T+1}^1=\bm{0}$; the interior-date formulation is the one directly relevant for the empirical implementation, which does not treat the final observed purchase period as terminal.
Consistency therefore requires the observed price vector $\bm{\rho}_t^+$ to lie in the column space of $\bm{\tilde{B}}_t$, yielding the following necessary condition.
Violation of (ref) at any single date is sufficient to reject the model, providing a sharp, low-dimensional falsification criterion that can be evaluated period by period. The restriction coincides with the necessary spanning condition under intertemporal separability in blow_revealed_2008: habit formation changes the level of shadow prices but does not expand the price manifold implied by the mapping from goods to characteristics. As such, (ref) is a structural constraint driven by the geometry of $\bm{A}$ rather than by the curvature or stability of preferences. Because (ref) is only necessary, however, it does not guarantee consistency: even when (ref) holds, the inequalities for goods consumed at zero quantities may still be infeasible. Nevertheless, (ref) substantially reduces the feasible set and provides a fast diagnostic for empirical implementation.
Beyond its falsification role, the rank restriction also has limited identification content. For any $t$, (ref) requires the observed price vector $\bm{\rho}_t^+$ to lie in the column space of $\bm{\tilde{B}}_t$. Because $\bm{B}_t^a$ is formed by rows of $\bm{B}_t$, the augmented matrix $\bm{\tilde{B}}_t$ has the same column space as $\bm{B}_t'$, so the feasible set of price vectors is at most $J$-dimensional. When habit formation is present ($J_2>0$), the mapping from $(\bm{\pi}_t^0,\bm{\pi}_{t+1}^1)$ to $\bm{\rho}_t^+$ is therefore not one-to-one: reallocating shadow value between contemporaneous and lag components along the habit-forming directions can leave $\bm{\rho}_t^+$ unchanged. Viewed through (ref) alone, the decomposition into contemporaneous and habit components is thus generically set-identified. In the running cereal example, prices may identify the overall shadow value of “nutty-ness” and “sensory intensity,” but not separately how much of the value of sensory intensity reflects current taste versus its continuation value through habits.
The full characterisation in Theorem (ref) can nevertheless sharpen this conclusion. Condition (ref) links the stacked discounted shadow prices $\tilde{\bm{\pi}}_t$ across dates through cyclical monotonicity evaluated at the observed augmented bundles $\tilde{\bm{z}}_t$, and therefore uses quantity variation as well as prices. These additional restrictions can shrink the admissible set of decompositions relative to the price-span argument embodied in (ref). For example, the data may satisfy Theorem (ref) under the dynamic model while violating the static restriction $\bm{\pi}_t^1\equiv \bm{0}$, in which case the zero-habit specification is excluded from the identified set. Absent further structure, however, Theorem (ref) still need not point-identify, or sign, the habit component separately from the contemporaneous one: (ref) disciplines the joint evolution of the stacked vectors $\tilde{\bm{\pi}}_t$, but does not in general undo the non-uniqueness in the decomposition induced by (ref).
I now relax the assumption that prices are observed for all market goods. In many applications, prices are recorded only for goods that are actually purchased, giving rise to a missing price problem.\footnote{Throughout, I assume that the technology matrix $\bm{A}$ is known and time-invariant. This reflects settings where characteristics can be directly observed or constructed (e.g., nutritional content, design features, emissions ratings), even when market prices are only recorded for purchased items; in practice, missing prices are far more common than missing characteristics.} Missing prices complicate RP analysis because imputing unobserved prices requires auxiliary assumptions. An alternative is to treat missing prices as unknowns and ask whether there exist values that render the data rationalisable. This existence approach should be understood as a partial-identification device: without further restrictions, one can always rationalise non-purchases by assigning prohibitively high unobserved prices, so the goal is to characterise when the observed prices alone already force a violation (or allow rationalisation) under the maintained technology. I now formalise this approach.
Let $\bm{\rho}^+_t$ denote the $K^+_t \leq K$ sub-vector of period-$t$ discounted prices for goods with strictly positive demand, and let $\bm{B}_t$ and $\bm{B}^a_t$ denote the corresponding $J \times K^+_t$ and $J_2 \times K^+_t$ sub-matrices of $\bm{A}$ and $\bm{A}^a$. Let $\bm{\rho}^0_t$, $\bm{B}^0_t$, and $\bm{B}^{a,0}_t$ denote the complementary sub-vectors and sub-matrices associated with zero demand. The full discounted price vector is $\bm{\rho}_t=(\bm{\rho}^+_t,\bm{\rho}^0_t)$. I can then state an Afriat-type characterisation for the habits-over-characteristics model with missing prices.
Proof: See Appendix (ref). $\qed$
Relative to Theorem (ref), the missing-price test conditions only on purchased-good prices and is therefore weaker. In empirical settings where the initial stock is unobserved and the observed sample window is not treated as the consumer's terminal horizon, the implementable counterpart is the observable interior-date analogue of Theorem (ref), which evaluates the restrictions on dates for which both the lagged bundle and the one-step-ahead continuation term are observed.
Before turning to the empirical application, I record two immediate corollaries that situate the habits-over-characteristics model within the RP literature. First, when characteristics coincide with market goods, the framework collapses to the habits-over-goods model of crawford_habits_2010. Second, under intertemporal separability and exponential discounting, it reduces to the characteristics-based model of blow_revealed_2008. Both results follow directly from Theorem (ref) once the model is specialised to the relevant limiting cases. Formal definitions and derivations for these special cases are provided in Appendix (ref).
When characteristics coincide with market goods---that is, when $J = K$ and the technology matrix satisfies $\bm{A} = \bm{I}_J$---the habits-over-characteristics framework reduces to a standard habits-over-goods model. In this case, the distinction between goods and characteristics disappears, and the intertemporal first-order conditions involve current and lagged consumption of habit-forming goods directly. To match crawford_habits_2010, I maintain the additional assumption that all goods are consumed in strictly positive quantities. Throughout this subsection, I adopt the terminal convention $\bm{\rho}_{T+1}^{a,1}\equiv \bm{0}$, so the final-period habit-good restriction has no continuation term. I also normalise the marginal utility of lifetime wealth to one, $\lambda=1$, without loss of generality.
Given this definition, I obtain the following result, equivalent to that in crawford_habits_2010 under the same normalisation.\footnote{Equivalence uses the normalisation of the marginal utility of lifetime wealth $\lambda = 1$, which can be imposed without loss of generality.}
Proof: See Appendix (ref). $\qed$
If habit formation is absent so that $\bm{z}_t = \bm{z}_t^c$ for all $t$, the model reduces to a characteristics-based framework with intertemporally separable preferences. Unlike blow_revealed_2008, who remain agnostic about intertemporal allocation, my formulation embeds this static characteristics model within a lifecycle problem with exponential discounting.\footnote{By a lifecycle problem I mean that the consumer chooses the entire consumption path to maximise lifetime utility subject to a single present-value budget constraint under exponential discounting.} The presence of a single intertemporal budget constraint implies a single shadow value of lifetime wealth, so observed choices must satisfy both intratemporal utility maximisation over characteristics and intertemporal optimality. Consistency with this lifecycle formulation is defined as follows. As in the rest of the paper, I normalise the marginal utility of lifetime wealth to one.
This recovers the hedonic pricing equation of gorman_possible_1956 and the first-order condition of blow_revealed_2008 when the static characteristics model is embedded in a lifecycle framework. From this, I obtain the following characterisation.
Proof: See Appendix (ref). $\qed$
Together, these corollaries clarify the paper's value added. The framework unifies existing RP results in goods space and in characteristics space within a single dynamic hedonic model, and it shows how the interpretation of prices changes once habit formation over attributes is allowed. Relative to intertemporal separability, the dynamic model introduces continuation values into the shadow-price system and delivers diagnostics---via feasibility of (ref)--(ref) and the rank condition (ref)---that separate failures of the hedonic price representation from failures of dynamically coherent preferences.
This section applies the framework to household-level cereal purchases to evaluate the empirical content of dynamic hedonic models in scanner data. Using these data, I show that most differences in raw rationalisability across models are driven by the structural restrictions imposed by the hedonic representation of observed prices, while allowing for habits systematically improves behavioural coherence conditional on those constraints. The goal of the application is to illustrate the framework in a realistic scanner setting, where only purchases are observed and the mapping from purchases to underlying consumption states must be treated as a maintained approximation rather than as a directly observed object.
The empirical exercise proceeds sequentially. First, I ask a structural question: do observed within-period price vectors admit a hedonic representation? Equivalently, do they satisfy the equalities implied by the goods-to-characteristics technology (i.e., (B2$^+$) in Theorem (ref)), so that characteristic shadow prices are well defined? These equalities act as a gatekeeper: if they fail, the behavioural test is not economically meaningful because the relevant shadow prices do not exist. Second, conditional on structural feasibility, I ask a behavioural question: do observed purchase sequences satisfy the dynamic RP inequalities (i.e., (B1$^+$)) when evaluated at the implied shadow prices? In the data, these inequalities are applied to purchase-period bundles, so the resulting exercise should be read as a diagnostic of dynamic coherence under the maintained approximation that purchases track the relevant underlying consumption state at the chosen aggregation.
The sequential decomposition yields two findings. First, most variation in raw pass rates when moving from goods space to characteristics space reflects the hedonic price-system restrictions rather than differences in behavioural fit; empirically, however, restoring hedonic consistency typically requires only modest price adjustments. Second, conditional on satisfying those structural restrictions, allowing for habits systematically improves behavioural coherence.
Market goods are UPC-level products and characteristics are their nutritional content and a small set of descriptive indicators. The key empirical challenge is that prices are observed only for purchased items. Accordingly, I implement the missing-price test of Section (ref).\footnote{Imputing prices for unpurchased goods is feasible (e.g., using regional price indices), but it introduces auxiliary assumptions that can blur whether failures reflect preferences or the imputation procedure.} In this setting, behavioural discipline is inherently limited by sparse price support and modest intertemporal budget variation, so the diagnostic provides a lower bound on the model's behavioural content. In environments with richer price and quantity support, the same framework may generate sharper behavioural restrictions.
I use household-level scanner data on cold cereal purchases from the IRI Academic Datasets' BehaviorScan panel. The panel covers two U.S. markets (Pittsfield, MA, and Eau Claire, WI) over 2010--2011, with purchases recorded at checkout via household ID cards and Universal Product Codes (UPCs). I restrict attention to static households that participate in all 12 months of a calendar year, so recruitment and attrition occur only at year-end.
I compute household-specific time periods to accommodate heterogeneity in purchase frequencies. For household $i$, let $S_i$ denote the span from first to last purchase and $G_i$ the longest interpurchase gap (including endpoints). I set $T_i=\lfloor S_i/G_i\rfloor$ and partition $S_i$ into $T_i$ equal-length bins, which guarantees at least one purchase in each period by construction. Households with $T_i<3$ are excluded to ensure at least two observed transitions in the one-lag model. This aggregation is conservative for a one-lag specification: it ensures that the lagged bundle is observed rather than imputed, hence dynamic restrictions are evaluated on realised purchase transitions. After additionally dropping households with purchases lacking characteristics information (described below), the analysis sample contains $N=2{,}282$ households.\footnote{Appendix (ref) documents the full sequence of sample construction and reports balance tests comparing households excluded due to missing characteristics data with the final analysis sample. Differences are modest in magnitude, suggesting limited scope for selection on observables.} Table (ref) summarises the resulting panel structure and the scale and variety of the implied choice problems.
Households purchase on average 7.9 units per period (median 6.0). Combined with a median period length of 98 days, this pattern is consistent with ongoing consumption rather than extreme purchase spikes. While promotions may induce some stockpiling, explaining the observed interpurchase gaps purely through inventory accumulation would require implausibly large stock build-ups relative to total quantities purchased. At the same time, within-period baskets remain narrow in UPC variety: the median household-period contains only two distinct purchased products, even though households typically buy several units. I therefore interpret each household-period as a meaningful dynamic choice observation, but not as a literal measure of contemporaneous nutrient intake: it accommodates idiosyncratic shopping frequencies while preserving the temporal structure required for testing whether aggregated purchase bundles behave in a manner consistent with the dynamic RP restrictions.\footnote{Similar coarse aggregation is common in empirical work on dynamic demand when purchase occasions are intermittent (e.g., crawford_habits_2010 uses quarterly panels in an application to tobacco). I obtain qualitatively similar cross-model comparisons under a common monthly aggregation (i.e., $T_i=24$ for all $i$), albeit in a much smaller effective sample because zero-purchase months become pervasive.}
Figure (ref) shows substantial heterogeneity in shopping frequency (median $T_i=6$; median period length 98 days). Within each period I aggregate UPC-level quantities and nominal expenditures and compute unit values as expenditure-to-quantity ratios.\footnote{Expenditures are recorded in nominal dollars and converted to present-value terms for the lifecycle formulation using a monthly interest-rate series (30-Year Fixed Rate Mortgage Average, FRED fred_mortgage30us); results are unchanged under nominal values given the short sample window.} Because unpurchased goods have no observed unit values, the test treats missing prices as unknowns and searches for completions consistent with rationalisability. Reported pass rates are therefore upper bounds: a household that fails cannot be rescued by any price imputation, while a household that passes does so under at least one completion.
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The model is defined over characteristics rather than market goods. For each UPC I construct a vector of nutritional and descriptive attributes by merging the IRI product file to the (now-defunct) NuVal shelf-labelling database and, where NuVal is missing, supplementing with data from the FatSecret Platform API, following the approach in barahona_equilibrium_2023. I standardise all nutrients to a 100g basis so that continuous characteristics are comparable across products. Since scanner quantities are recorded in package counts, the resulting characteristics matrix should be interpreted as a linear summary of purchased product attributes. To handle IRI's placeholder codes for some private-label items, I first map System-88 pseudo-UPCs to their corresponding real UPCs prior to merging. The merged characteristics cover over 97% of purchase-weighted observations; I drop unmatched purchase records and then drop households with any remaining unmatched purchases, so every household in the analysis sample has a complete characteristics mapping for all of its observed purchases.
I construct $J=23$ characteristics. Eight are continuous Nutrition Facts Panel measures (calories, carbohydrates, total fat, saturated fat, fibre, protein, sodium, and sugar). The remaining 15 are binary indicators capturing salient ingredients and descriptors (10 indicators) and the five most prevalent brands (Kellogg's, General Mills, Post, Quaker, and Kashi). Table (ref) reports the binary definitions. In the baseline specification I treat sugar and sodium as habit-forming characteristics ($J_2=2$) and take the remaining characteristics as non-habit-forming, motivated by evidence that sugar and salt may activate reward pathways in ways analogous to addictive substances avena_evidence_2008, cocores_salted_2009. Section (ref) reports robustness to alternative partitions. Additional descriptive evidence on purchase intensity, brand concentration, and the distributions of prices and characteristics is reported in Appendix (ref).
Two patterns emerge in the raw rationalisability outcomes: allowing for habits increases pass rates within characteristics space, and goods-based representations exhibit substantially higher pass rates. As the sequential logic of the paper makes clear, however, these raw differences conflate structural and behavioural components. Section (ref) decomposes these margins and quantifies the severity of violations.
The test is implemented at the household level, allowing full heterogeneity in pass/fail outcomes and in the shadow-price structure of rationalisable households. Because habits enter with a one-period lag, the empirical exercise uses the observable interior-date analogue of Theorem (ref). Structural equalities are imposed on $t=2,\ldots,T-1$, since period $t=1$ depends on the unobserved initial stock and I do not treat the last observed purchase period as the consumer's terminal horizon. The behavioural Afriat inequalities are then evaluated on the retained dates $t=2,\ldots,T$, so the final observed period continues to discipline the shadow-price sequence. Unless otherwise specified, I impose a lifecycle model with a single intertemporal budget constraint, so the marginal utility of income is constant across periods.
Under the baseline habits-over-characteristics specification, 1{,}248 of 2{,}282 households (54.69%) satisfy the test. Varying the set of habit-forming characteristics has essentially no effect on classification: restricting habits to sugar alone (54.60%), sodium alone (54.65%), or allowing all 23 characteristics to be habit-forming (54.69%) changes the pass rate by at most 0.09 percentage points and reclassifies no more than two households. In this application, the data therefore do not isolate sugar or sodium as uniquely responsible for the dynamic improvement; rather, the main empirical action comes from the maintained hedonic representation, with habits mattering at the behavioural margin once that representation is imposed. Table (ref) reports the full set of specifications.
Raw pass rates differ sharply across representations (Table (ref)). Moving from characteristics to goods sharply increases rationalisability from 54.7% to 99.6% under dynamic preferences. This large difference is driven by the much greater dimensional flexibility of goods-based representations, which impose no cross-good price restrictions. In this application, the structural restrictions bite through the geometry of the active purchased bundle rather than through broad within-period variety. Many of the apparent “failures” in characteristics space nevertheless correspond to economically small deviations from those hedonic restrictions. Section (ref) makes this precise by separating structural and behavioural sources of empirical discipline and quantifying the magnitude of violations.
By contrast, removing habits within characteristics space reduces the pass rate from 54.7% to 52.4%. While this difference in levels is modest, the paired comparisons below show that the reclassification is entirely directional, with households failing under static preferences but passing once dynamics are introduced. I show later that allowing for habits also reduces the severity of behavioural violations on average, but the empirical gains are concentrated rather than universal.
The reclassification pattern is strongly directional. Removing habits from the baseline characteristics model reclassifies 53 households, all of whom fail under static preferences but pass once intertemporal dependence is introduced ($\text{p-value}<10^{-15}$). By contrast, changing the allocation of habit-forming characteristics reclassifies at most three households and yields no statistically significant differences. Table (ref) reports the full set of paired comparisons.
Comparisons with goods-based models reveal even larger directional differences. In these cases, most switching households fail the characteristics-based test but pass the corresponding goods-based alternative. As shown in the next subsection, this pattern reflects the much greater dimensional flexibility of the goods representation, which imposes no cross-good price restrictions, rather than tighter behavioural alignment.
Taken together, these results indicate that while the precise allocation of habits across characteristics plays little role in classification, introducing intertemporal dependence in characteristics space improves rationalisability for a limited but directionally important subset of households. In this application, the main empirical action comes from the maintained hedonic representation, while habits matter at the behavioural margin. The results are therefore better read as illustrating the geometric consequences of the hedonic representation than as isolating particular attributes as uniquely responsible for dynamic behaviour.
Raw pass rates alone are insufficient to interpret fit: they conflate empirical success with permissiveness and provide only a binary measure of failure. As emphasised by selten_properties_1991, a model may rationalise many datasets either because it captures economically meaningful structure or because it imposes few substantive restrictions on observable outcomes. Moreover, a binary pass/fail outcome does not reveal how severe a violation is when the model fails. I therefore separate rationalisability into structural and behavioural components and quantify violations on each margin using continuous discrepancy measures. This approach follows a broader methodological insight that representation theorems naturally induce continuous measures of rationality violations, since their axioms hold if and only if a rationalising object exists (e.g., andrews2026revealed).
In the hedonic setting, this distinction admits a natural decomposition. Rationalisability in the habits-over-characteristics model requires joint satisfaction of two conceptually distinct restrictions. Structural equalities (B2$^{+}$) link observed prices to product characteristics through the hedonic technology and act as overidentifying restrictions on the admissible shadow-price representation. This is the gatekeeper stage of the empirical analysis. Behavioural inequalities (B1$^{+}$) constrain intertemporal choice through shadow prices and the discount factor (Theorem (ref)). Separating these margins clarifies where empirical discipline originates in characteristics-based valuation and how it differs from more flexible goods-based representations.
The structural equalities require that, in each household-period, the observed price vector $\bm{\rho}_{t}^{+}$ lies in the column space of the augmented characteristics matrix $\bm{\tilde{B}}_t :=
$. In principle this is a dimensionality-reduction restriction: prices must be representable through a lower-dimensional characteristics technology rather than arbitrary goods-specific shifters. In the scanner environment studied here, however, the active choice sets are typically narrow, so the empirical bite of the restriction depends on the realised rank of the purchased bundle. When the purchased-good matrix has full row rank, the column space of $\bm{\tilde{B}}_t$ spans all of $\mathbb{R}^{K_t^+}$ and the structural equalities are mechanically satisfied. Non-zero structural distances therefore arise precisely in those household-periods where the active bundle is rank deficient, either because the household buys too few linearly independent products or because the purchased UPCs have highly similar characteristic profiles.
This is empirically plausible in cereal data, where closely related UPCs often differ only in package size, branding, or minor formulation details and therefore carry very similar measured characteristic vectors. Even after conditioning on a rich set of nutritional and descriptive characteristics, scanner prices also reflect retailer pricing strategies, temporary promotions, and mark-ups driven by market power or shelf placement that do not correspond to attributes households consume. When the active bundle is low rank, those retailer-specific price components cannot be absorbed by the hedonic system and therefore appear as structural residuals.
To quantify the severity of these violations, I compute for each household-period the Euclidean distance \[ d_t = \big\| \bm{\rho}_{t}^{+} - \bm{\tilde{B}}_t \bm{\tilde{B}}_t^{+} \bm{\rho}_{t}^{+} \big\|, \] where $\bm{\tilde{B}}_t^{+}$ denotes the Moore--Penrose pseudoinverse and $\bm{\tilde{B}}_t \bm{\tilde{B}}_t^{+}$ is the orthogonal projector onto the equality manifold implied by the hedonic representation. Equivalently, observed prices admit the orthogonal decomposition \[ \bm{\rho}_t^{+} \;=\; \bm{\tilde{B}}_t \hat{\bm{\pi}}_t \;+\; \bm{r}_t, \] where $\hat{\bm{\pi}}_t$ minimises $\|\bm{\rho}_t^{+} - \bm{\tilde{B}}_t \bm{\pi}\|$ and $\bm{r}_t$ is orthogonal to the column space of $\bm{\tilde{B}}_t$. The distance $d_t = \|\bm{r}_t\|$, measured in dollars, therefore captures the minimal joint price adjustment required for a hedonic price representation to exist. For example, $d_t = 1.0$ means that the minimum adjustment vector to prices has Euclidean norm 1 dollar; equivalently, the sum of squared price adjustments across the goods purchased that period is 1.
One response to violations of the hedonic equalities is to augment the technology with latent characteristics, as in blow_revealed_2008, thereby expanding the price manifold until the equalities hold. I do not pursue this route here. My objective is not to restore rationalisability by construction, but to quantify how demanding a given hedonic representation is in the data and to separate structural misspecification from behavioural inconsistency.
Figure (ref) plots the distribution of mean distances across households for both characteristics- and goods-based specifications. The characteristics models exhibit a wide, right-skewed distribution centred above zero. In this application, those distances should be read as evidence that many active bundles fail to span their own purchased-good price space once prices are projected through the maintained characteristics technology. Even so, the implied unit-price violations are often modest in magnitude: most household-periods require a minimum joint price adjustment with Euclidean norm below 1 dollar to restore hedonic consistency. As a rough benchmark, average household cereal expenditure is \$22.1 per period. By contrast, goods-based models impose no cross-good price restrictions and therefore satisfy the structural equalities mechanically, yielding $d_t = 0$.
An important implication of this geometry is that all characteristics-based specifications impose identical structural restrictions. Habit-forming characteristics enter $\bm{\tilde{B}}_t$ only as duplicated rows of $\bm{B}_t$ and therefore neither increase its rank nor enlarge the feasible price set. Structural restrictiveness is thus governed entirely by the hedonic representation itself, not by the allocation of habits across characteristics. In the present data, this means that structural rejection is best understood as a statement about the linear algebra of narrow purchased bundles---their rank and characteristic collinearity---rather than about the dynamic specification per se. Moreover, these structural restrictions operate at the level of prices and characteristics and are conceptually distinct from inventory dynamics: even if households smooth consumption through inventories, prices must still admit a hedonic representation for shadow prices to be well defined.
To measure behavioural restrictiveness, I adapt the Critical Cost Efficiency Index (CCEI) following afriat1973system and varian_goodness--fit_1990. The CCEI measures the smallest proportional relaxation of revealed affordability required for the behavioural inequalities in (B1$^{+}$) to admit a solution.\footnote{The CCEI is interpreted here strictly as a measure of RP slack, following its original cost-efficiency interpretation in afriat1973system. As emphasized by echenique_meaning_2022, it should not be interpreted as a welfare loss or a measure of foregone surplus.} Economically, a CCEI of $\eta$ indicates that the revealed-affordability comparisons in the lifecycle RP test need only be relaxed by at most $(1-\eta)\times 100\%$ for the observed intertemporal choice path to become rationalisable. Values close to one therefore indicate that behaviour is nearly dynamically rational, while lower values signal more severe violations. The index is defined only for households whose prices lie exactly on the equality manifold, because characteristic shadow prices---and hence the behavioural inequalities themselves---are well defined only when the hedonic equalities hold. In this sense, the CCEI plays an analogous role to a goodness-of-fit statistic for intertemporal choice: it quantifies how much slack must be introduced before the model can rationalise observed behaviour, holding the price system fixed.
Figure (ref) shows that behavioural violations are generally modest, but systematically smaller when habits are allowed. All characteristics-based specifications exhibit substantial mass near unity, indicating that once the hedonic structure is satisfied, only limited perturbations are needed to rationalise behaviour. However, the static characteristics model without habits displays a thicker lower tail, reflecting greater intertemporal inconsistency even when structural feasibility holds. A parallel pattern arises in the goods domain: the habits-over-goods specification yields CCEIs tightly concentrated near one, while the static goods model exhibits a wider distribution with more mass below 0.95. These patterns indicate that habit formation strengthens behavioural discipline by reducing intertemporal reversals. This matters for interpretation: habits change the mapping from observed prices to marginal valuations by introducing the dynamic wedge characterised in Section (ref).
While this decomposition clarifies where empirical discipline originates within the model, it does not by itself establish how informative these restrictions are relative to plausible alternatives in the empirical environment. I therefore complement it with a simulation benchmark in the spirit of fudenberg_how_2023, which evaluates how unusually well the model fits the observed data relative to nearby feasible perturbations of the scanner environment. The benchmark preserves each household's zero pattern and total expenditure while allowing prices and quantities to vary locally, thereby inducing a comparison distribution over structural and behavioural discrepancy measures. Full details are provided in Appendix (ref).
On the structural margin, the benchmark confirms that the hedonic equalities impose substantial empirical discipline. Observed scanner prices are, on average, closer to the equality manifold than locally perturbed price systems, yet are rarely extreme outliers relative to the induced comparison distribution. This indicates that the hedonic restrictions are demanding and not trivially satisfied. By contrast, goods-based models impose no cross-good price restrictions and therefore cannot fail on the structural margin, accounting for their much higher raw pass rates.
On the behavioural margin, CCEI values are extremely close to one in both the observed and locally perturbed data, leaving little scope for behaviour to appear unusually efficient in a quantile sense. This reflects the empirical environment---sparse active choice sets and limited intertemporal budget variation---rather than a lack of behavioural content in the model.
Taken together, the price geometry, behavioural slack measures, and simulation benchmark clarify the interpretation of raw pass rates. Structural restrictions embedded in the hedonic representation largely determine differences across goods and characteristics models, but in this application they do so through the rank properties of the active purchased bundles rather than through a broad market-level dimension count alone. Conditional on those restrictions, allowing for habit formation systematically improves intertemporal coherence relative to static preferences. Goods-based models achieve high pass rates primarily because they impose little structural content, not because they deliver a tighter account of behaviour.
The RP conditions are evaluated conditional on the discount factor $\beta$. Appendix (ref) reports the fraction of rationalisable households consistent with each value on a fine grid $\beta \in [0.95,1]$. Acceptance probabilities are uniformly high across the grid under both habits-over-characteristics and habits-over-goods specifications, with no economically meaningful monotonic pattern.
These results indicate that, conditional on the hedonic price restrictions, the behavioural inequalities impose only weak discipline on intertemporal discounting in this environment. The identification sets for $\beta$ are wide, cautioning against interpreting discount factors recovered from nonparametric dynamic RP tests as tightly identified structural parameters in scanner data.
Pass/fail outcomes are primarily associated with the scale and complexity of the observed choice problem rather than with demographics per se. Table (ref) reports average marginal effects from probit regressions of the pass indicator on household characteristics. Column (1) includes demographics only; Column (2) adds measures of purchasing intensity and variety.
In the demographic-only specification, households with children are 10.6 percentage points less likely to pass. Once purchasing controls are added, this effect attenuates and becomes statistically insignificant. By contrast, scale measures remain economically large and significant: an interquartile increase in two-year cereal expenditure is associated with roughly a 25 percentage point lower probability of passing, while a comparable increase in product variety reduces the probability by about 8 percentage points.
Taken together, these results suggest that household composition matters primarily through the scale and complexity of the observed choice problem. Larger and more diverse purchasing patterns generate a greater number of structural and behavioural constraints, increasing the likelihood that at least one is violated. For instance, when a household purchases products with sharply different nutritional profiles across periods, the implied cross-attribute shadow-price system must rationalise a wider range of price--quantity trade-offs than in households that repeatedly buy a narrow set of similar cereals. Demographics per se have limited explanatory power once this effective dimensionality is accounted for. The demographics-only specification has almost no explanatory power (McFadden pseudo-$R^{2}=0.009$), whereas the specification with purchasing controls reaches a more substantial 0.176.
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This paper provides a nonparametric foundation for dynamic hedonic valuation. I characterise when observed prices and choices admit a coherent life-cycle interpretation in which utility depends on current characteristics and lagged consumption of a habit-forming subset. The main result is an Afriat-type theorem in characteristics space, together with a missing-price extension suited to scanner environments.
The framework sharpens the empirical content of hedonic valuation in three ways. First, it shows how habits alter the interpretation of hedonic prices by introducing a continuation-value wedge. Second, it separates two distinct sources of discipline: structural equalities that restrict the admissible shadow-price representation through the maintained characteristics technology and behavioural inequalities that restrict intertemporal choice conditional on that technology. Third, it provides quantitative diagnostics that distinguish model rejection from the severity of the underlying violation.
The central implication is that dimensionality reduction and dynamics play different roles. Moving from goods to characteristics yields parsimony but imposes geometric discipline on the admissible representation of observed prices. Habits do not relax that discipline. Instead, they change the interpretation of prices within the feasible hedonic system by allowing current consumption to affect future marginal valuations. Dynamic hedonic valuation is therefore relevant only after a coherent characteristics representation exists. If the price-spanning restriction fails, the inconsistency originates in the hedonic representation itself and no dynamic extension can restore consistency. If the structural restrictions hold, the behavioural test isolates whether time separability is the binding assumption and whether a dynamic interpretation is additionally admissible relative to static valuation.
The cereal application illustrates this logic in a purchase-based scanner setting. Goods-based benchmarks pass at very high rates, while characteristics-based models fail much more often because observed prices frequently violate the hedonic spanning restriction. Yet the associated distance measures show that those structural violations are often economically modest. Conditional on structural admissibility, behaviour is close to dynamically rational, and allowing for habit formation improves behavioural coherence relative to static characteristics models for a subset of households.
Several directions for future work follow directly from the framework. On the empirical side, richer data with denser price support, more frequent observation, and greater intertemporal budget variation would sharpen both the structural and behavioural content of the test. On the modelling side, it would be useful to extend the analysis to stochastic choice, to settings with unobserved heterogeneity beyond household-by-household rationalisability, and to mixed environments in which utility depends on current goods while habits attach only to a subset of lagged characteristics. More generally, the framework is best understood as a sharp diagnostic of internal coherence rather than as a structural estimator of primitives: rationalisability does not, by itself, identify a unique dynamic preference representation or uniquely attribute failure to a single underlying mechanism.
Taken together, the results provide a disciplined benchmark for dynamic hedonic modelling. By separating structural from behavioural restrictions, the framework clarifies when a dynamic hedonic interpretation matters for valuation and when apparent failures instead reflect misspecification of the price-characteristics mapping. It therefore provides a theory-grounded and empirically implementable benchmark for distinguishing behavioural departures from structural misspecification in dynamic hedonic environments.