A Behavioral Foundation of Satiation and Habituation

Abstract

Tastes change over time. People’s tastes are distorted through two channels: satiation formation and habit formation. In this paper, we develop a theoretical foundation of satiation and habituation by an axiomatic approach. Our theory is based on a hierarchy of preference conditions called compensation independence. The behavioral assumption underlying the preference conditions are the psychological compensation of human beings. I flesh out an axiomatic system for general models of satiation and habit formation, which contains many functional forms in the literature as special cases. Moreover, I advance the axiomatization to accommodate the linear representations of satiation and habit formation that are prevailing in the literature. This paper contributes to the birth of a new generation of the behavioral foundation for modeling satiation and habit formation, which might improve on the current state of the art in understanding people’s tastes over time and preferences. Theoretically, this study contributes to the vein of time-nonseparable preferences.

1. Introduction

Consumer preference theories assume a fixed taste of people. However, the taste changing over time is an acknowledged fact. It motivates the development of habit models in the economic literature, e.g., Becker and Murphy’s rational addition model [1], Wathieu’s habituation model [2,3], and Gul and Pesendorfer’s harmful addiction model [4]. Rozen [5] proposes an axiomatization for intrinsic habit formation in which a habit compensation axiom is developed under the ordinal preference assumption. Models of habit formation can be further found in [6,7,8,9,10,11]. In this paper we use the terms “habituation” and “habit formation” interchangeably. Wathieu ([3], p. 588) advocates a distinct feature in two similar terms. Habit formation, mostly used by economists, assumes adjacent complementarity over time that predicts “the more you get, the more you want”. Representatives include [1,5,10,11]. Habituation, mostly used in management scientists, e.g., [2,3,12], assumes adjacent substitutability over time that predicts “the more you get, the less you want”. Compared with habit formation in an economic context, habituation exhibits richer implications in modeling customer behaviors, particularly after incorporating diminishing sensitivity, a property of prospect theory’s “gain-loss” value function [13]. This paper is closer to the implication of habituation.

The recent management literature sheds light that satiation is another aspect that distorts consumers’ taste across periods. Baucells and Sarin ([14]; BS07 henceforth) develop a general model of satiation (SA) using a modification of the discounted utility (DU). Their model says that consumer’s satiation that is accumulated from past consumptions can produce boredom of consumptions in the future. It predicts that satiation decreases the marginal utility of future consumption. McAlister [15] proposes an earlier model of satiation that says that satiation can be an ideal level of consumption inventory over an attribute. Baucells and Sarin ([12]; BS10 henceforth) further develop a hybridization of satiation and habituation (SH) by synthesizing their SA model and Wathieu’s [2] habituation (HA) model. The DU assumes the time independence thereby the period utility is calculated afresh. The total utility $U(.)$ is the sum of the period utility ${\mathcal{U}}_t(.)$, given as $U(.)={\displaystyle \sum }_{t=1}^T{\mathcal{U}}_t(.)$. Assuming time-separable preferences, ${\mathcal{U}}_t(.)$ can be materialized by the general consumption utility $u_t\left(c_t\right)$ pertaining to the t-period consumption level $c_t$. Using ${\stackrel{\rightharpoonup }{c}}_{1,T}$ to denote the consumption sequence $\left(c_1,c_2,\dots ,c_T\right)$, the total utility can be represented as $U\left({\stackrel{\rightharpoonup }{c}}_{1,T}\right)={\displaystyle \sum }_{t=1}^T{\delta }^t{u}_t\left(c_t\right)$. The DU model that is initially proposed by Samuelson [16] and axiomatized by Koopmans [17] has been the leading model of intertemporal choices for many years. However, its independence axiom limits the descriptive capacity of DU. As evidenced by the literature, preferences over the current consumptions could be influenced by the past consumptions through two distinct channels: habit formation and satiation formation [18]. Allowing time-nonseparable preferences by relaxing the independence axiom of DU has been widely accepted in the literature. For example, Bleichrodt et al. [19] relax Koopmans’ preference conditions and thus further accommodate more flexible utility functions.

The literature in economics has exhibited a preference-dependent formula in modeling habit formation. The period utility represents ${\mathcal{U}}_t\left(c_t\right)=u_t\left(c_t-h_t\right)$ for $t=1,\dots ,T$ where the habituation function (HF) $h_t=h\left({\stackrel{\rightharpoonup }{c}}_{1,t-1}\right)$. Meanwhile, the literature in management exhibited an additive utility structure in modeling satiation formation. The period utility is given by ${\mathcal{U}}_t\left(c_t\right)=u_t\left(s_t+c_t\right)-u_t\left(s_t\right)$ where the satiation function (SF) $s_t=s\left({\stackrel{\rightharpoonup }{c}}_{1,t-1}\right)$. Both HF and SF depend on the time stream of prior consumptions ${\stackrel{\rightharpoonup }{c}}_{1,t-1}$. By hybridizing HA and SA, BS10’s SH model formulates that the period utility represents ${\mathcal{U}}_t\left(c_t\right)=u_t\left(s_t-h_t+c_t\right)-u_t\left(s_t\right)$ where HF $h_t$ and SF $s_t$ pertaining to t. Period utility ${\mathcal{U}}_t\left(c_t\right)$ is interpreted as the experienced utility [20]. In this paper, we first provide the axiomatization for general models of satiation and habit formation, which contains the SA, HA, and SH models as special cases. Furthermore, in literature, linearity for HF and SF are prevailing; for example, a linear HF in [5] and [2] and a linear SF in BS07 and BS10. Therefore, the second work reported in this paper is to advance axiomatization for accommodating the linear representations of satiation and habit formation.

The present theory is premised on the notion of psychological compensation in human beings. This behavioral assumption is fundamental and intuitive. For a simple understanding, a busy father in the past may tend to compensate his daughter by spending more time with her in the future. The lack of sleep on weekdays because of the heavy workload could be balanced to an extent by compensating the time of sleep on the weekend. A hungry man may tend to overeat once having a rich meal. The psychology of compensation here is akin to the notion of craving. A poor man tends to spend money excessively once he becomes rich suddenly. A frequently partying student may be willing to party compensatorily after a period of confinement for exams. In this paper, we adopt the narrow sense of compensation, which is that people have a psychological reaction of compensation when consuming goods across periods. This idea can be materialized as a process of compensation, in which the decision maker’s (DM) preference over the consumption sequence is indifferent between this process before and after. This paper fleshes out that this behavioral assumption can be applicable for modeling intertemporal preference.

We put forward a hierarchy of preference conditions called compensation independence under the behavioral assumption of psychological compensation. Assuming cardinal preference over sequences of consumptions, compensation independence can arise: (i) the additive utility formula in the satiation; (ii) the preference-dependent utility formula in the habituation context; and (iii) the joint preference-dependent and additive utility formula in both satiation and habituation context. Inspired by the neutral continuation axiom of [21], where they use a well-being model, preference indifferences can be established in the sequences that have different lengths. Preferences can thus be defined on a set of sequences of consumption such as $\left\{c_1, {\stackrel{\rightharpoonup }{c}}_{1,2}, \dots , {\stackrel{\rightharpoonup }{c}}_{1,T}\right\}$. Period utility at t can be represented as ${\mathcal{U}}_t\left(c_t\right)=U\left({\stackrel{\rightharpoonup }{c}}_{1,t}\right)-U\left({\stackrel{\rightharpoonup }{c}}_{1,t-1}\right)$ for $t\ge 2$ and ${\mathcal{U}}_1\left(c_1\right)=U\left(c_1\right)$ for $t=1$, where $U(.)$ is defined as a cardinal preference over sequences. The additive structure derives from the difference in two cardinal utilities pertaining to the finite-and-different-length sequences. Thereby, we establish two specifications of the primitive independence condition, which are satiation compensation independence and habit compensation independence as detailed later.

Although both Rozen [5] and ourselves use the term “habit compensation” in the habituation context, the fundamental differences exist. Rozen’s theory considers the infinite (i.e., identical) length of sequences; therefore, the neutral continuation axiom adopted in this paper is invalid for hers. Rozen’s theory assumes ordinal preference, whereas this paper assumes cardinal preference. Rozen considers an infinite horizon in the axiomatization of intrinsic habit formation, whereas we advocate the necessity of considering a finite time horizon in the habituation context as justified by Wathieu ([2], p. 1555). Looking at the origin of the idea, Rozen’s habit compensation axiom is akin to Hicksian wealth compensation ([22]; see also Rozen [5], p. 1347), whereas the habit compensation in this paper comes from the behavioral compensation of human beings that matters in individual psychology.

Compared with the rich theoretical studies in habit formation, the axiomatizations of satiation are relatively rare [23]. He et al. ([24]; HDB henceforth) provides the first attempt to axiomatize the hybrid SH model. They assume the existence of a measurable value function under the value-utility framework [25] that accommodates the comparison of preference intensity over exchanges of outcomes. The preference condition in HDB, shifted difference independence, is a variant of Dyer and Sarin’s difference independence condition. This pioneering work well accommodates the existing utility formulas of satiation and habit formation. However, their auxiliary preference condition seems excessively strict and thus in a way is not intuitive enough. This paper attempts to relax such strict conditions and parsimoniously uses only preference as the primitive for a new axiomatic system of satiation and habit formation.

We use the example for a simple illustration. Given a two-period consumption stream ($c_1,c_2)\in C_1\times C_2$ where $C_1≔ℝ$ and $C_2≔ℝ$. We consider two exchanges: from (0, 2) to (0, 3) and from (1, 2) to (1, 3). Following HDB, the first exchange can balance the second exchange by adding a quantity of consumption $\Delta \left(0,1\right)$, from the peirod 1 to the period 2, which says that the exchange is from (1, 2+$\Delta \left(0,1\right)$) to (1, 3+$\Delta \left(0,1\right)$). The assumption of an existing measurable value function is the necessary condition of existing $\Delta \left(0,1\right)$. Following our theory, one just needs to consider the stream (1, 2) and verify the existence of $\Delta \left(0,1\right)$, such that the indifference (1, 2)~(0, 2+$\Delta \left(0,1\right)$) can be satisfied. The obtained indifference is through a compensation from period 1 to period 2. Period 2 is compensated by $\Delta \left(0, 1\right)$, which is triggered by a withdrawal of consumption in period 1.

Put differently, the effect of withdrawing the past consumptions until the zeroes can be assimilated through compensating a quantity $\Delta \left(0, 1\right)$ on a particular point of time subsequently, such that the preference is invariant across periods of consumptions. This process is underlain by assuming a proper preference condition as the compensation independences proposed in this paper.

Different from HDB’s theory, there is no need to assume a measurable value function in our theory. Unlike HDB, who impose to shift the value function for comparing the preference intensity over consumption streams, our theory only applies preference that must be underlain by the assumption of behavioral compensation in individual psychology. Our behavioral assumption could be more intuitive and generalized than HDB’s. As pointed out by Peter Wakker, the strength of preference is another way of saying preference intensity. For related literature, one can see Chai et al.’s theoretical treatment [26], “axiomatizations that use the strength of preference as primitive are less appreciated than axiomatizations that only use preference”.

The notion of preference intensity can also be used for interpreting our axiomatization. For example, the exchange from (1, 2) to (0, 2) balances the exchange from (1, 2) to (1, 2+$\Delta \left(0,1\right)$). Assuming a measurable preference order as a binary relation ${\succcurlyeq }^{*}$ defined on ($C_1\times C_2)\times (C_1\times C_2$), we derive the indifference $\left(0, 2\right)\left(1, 2\right){~}^{*}\left(1, 2+\Delta \left(0,1\right)\right)\left(1, 2\right)$. It says that the preference intensity of a withdrawal of consumption in period 1 equals the preference intensity of exchanging from the original stream $\left(1, 2\right)$ to the new stream $\left(1, 2+\Delta \left(0,1\right)\right)$ after the process of compensation. We only entail the indifference preference “${~}^{*}$”, whereas HDB entails entire binary relations of preference “${\succcurlyeq }^{*}$ (and ${\preccurlyeq }^{*}, {\succ }^{*},{\prec }^{*},{~}^{*}$)”. To derive an additive utility structure that accommodates the satiation context under “${\succcurlyeq }^{*}$”, HDB must impose a variation in the Dyer-Sarin independence condition because it underlies an additive multiattribute value function. Differently, the additive structure in our theory comes from the difference between two different-length-sequence-contingent cardinal utilities. We thus can only use preferences rather than preference intensity as the primitive of the axiomatization.

We organize this paper as follows: Section 2 presents the framework. Section 3 exhibits the behavioral assumption and the primitive preference conditions. In Section 4 and Section 5 we introduce the main preference conditions in the satiation and habit-based satiation contexts, respectively. Section 6 presents the axiomatic system for general SH models. Section 7 and Section 8 advance the axiomatizations for accommodating linear representations of satiation and habituation. We conclude this paper in Section 9. The Supplementary Materials provides all proofs.

2. The Framework

Suppose that a good is consumed in each period $t\in \mathbb{N}=\left\{1,\in iT\right\}$ from the set $C≔ℝ$. A consumption level $c_t\in C$ may be interpreted as a decision of consuming a quantity of the good. The sequence of consumptions that lasts from period $1$ to period $T$ is denoted as ${\overrightarrow{c}}_{1,T}\in {\stackrel{\rightharpoonup }{C}}_{1,T}≔{\times }_{t=1}^T{C}_t$. DM’s preference is defined on the binary relations $\succcurlyeq$ (and $\preccurlyeq , \succ ,\prec ,~$) over a finite discrete time horizon $t\in \left\{1,\dots ,T\right\}$. The DM has an intrinsic, outcome-contingent consumption utility $u_t\left(c_t\right)$ pertaining to $t$ that has been typically studied in economics. We assume that this utility accommodates the well-known S-shaped utility form [13]. It entails: (i) the concavity in the positive zone; (ii) the convexity in the negative zone; and (iii) a steeper loss than gain limb, or some other forms of loss aversion.

We assume that DM’s preferences over sequences of consumptions are cardinal. The desirability of a sequence can be represented by a cardinal utility $U(.)$. The preference relation $\succcurlyeq$ given on $U(.)$ is standardized by a weak order (complete, transitive, and continuity, Note that Kothiyal et al. [27] argue that the condition of continuity is not necessary for additive or average utility. A preference function represents $\succcurlyeq$ if $U(.)\to ℝ$ and $x\succcurlyeq y\iff U\left(x\right)\ge U\left(y\right)$. The $U(.)$ is defined on a set of sequences with a variable length from $t=1$ to $t=T$; the prospects $x$ and $y$ are from the set of sequences $\left\{{\stackrel{\rightharpoonup }{c}}_1, {\stackrel{\rightharpoonup }{c}}_{1,2}, \dots , {\stackrel{\rightharpoonup }{c}}_{1,T}\right\}$. We are interested in an additive representation of $U(.)$ that is a sum of the period utility ${\mathcal{U}}_t(.)$, as denoted $U\left({\stackrel{\rightharpoonup }{c}}_{1,T}\right)={\displaystyle \sum }_{t=1}^T{\mathcal{U}}_t\left(c_t\right)$.

Our axiomatization is choice-based under riskless preferences. If considered as the decision utility, ${\mathcal{U}}_t\left(c_t\right)$ is based on revealed preferences on a choice of the good. Maximization of the total utility obeys all classical interpretations in economics. If considered as the experienced utility, ${\mathcal{U}}_t\left(c_t\right)$ can be interpreted as a measure of hedonic sensations of consuming a good. Total utility is interpreted as a measure of happiness and individual well-being. Our framework can be extended immediately to lotteries over sequences of consumptions by imposing the axioms of von Neumann and Morgenstern’s expected utility [28]. We thus do not need another package of axioms for accommodating risky preferences.

We clarify the state space used in this paper. The large-horizon sequence of consumption is denoted as ${\overrightarrow{c}}_{1,T}\in {\stackrel{\rightharpoonup }{C}}_{1,T}$. This last period T can be naturally interpreted as the end of life. One can understand the per-period as a year in life, e.g., Deaton’s life-cycle theory [29]. We are thinking of the possibility when consuming a certain good is in a time window, such that satiation and habituation models account for the consumption during this time span like BS07 (p. 179).

We are interested in the sequence ${\overrightarrow{c}}_{s,k}$ for ${\overrightarrow{c}}_{s,k}\in {\overrightarrow{C}}_{s,k}$ where $1\le s\le t<k\le T$. We particularly focus on the consumption space ${\overrightarrow{C}}_{s,t}\times C_k$ for $s\le t<k$ and $k\in \left\{t+1,\dots ,T\right\}$. Without a special statement, we use $k$ for the present time. Given $t=k-1$, the state space is degenerated as $\left\{s,\dots ,t,k\right\}$. In our theorems, we use the space $\left\{1,\dots ,k\right\}$ by letting $s=1$. Please distinguish the subspace $\left\{1,\dots ,k\right\}$ that represents $\left\{s,\dots ,t,k\right\}$ and the life-long horizon $\left\{1,\dots ,T\right\}$. For generality, we tend to apply ${\stackrel{\rightharpoonup }{C}}_{1,T}$ where $1\le s\le t<k\le T$ in our axioms; for simplicity, we tend to apply ${\overrightarrow{C}}_{s,t}\times C_k$ or ${\overrightarrow{C}}_{1,k}≔{\times }_{i=1}^k{C}_i$ in our theorems.

3. Compensation Independence

We apply the narrow sense of compensation in individual psychology as the behavioral assumption. It says that the changes in past consumptions can be compensated to a certain extent by changing the consumption level at a particular point of time in the future. Once DMs identify this assumption, we say that there exists a compensating association between the past dimension that we call the reference point (RP), and the particular future dimension that we call the compensating point (CP). The compensating association is directional over time, from the RP to the CP.

Based on this behavioral assumption, we put forward the primitive preference condition as follows. Considering our running example, the lack of time of your weekdays’ sleeping maybe “exactly”, instead of “to an extent”, balanced by a compensatory time of your weekend’s sleeping. The “exactly balanced” means that your desirability on your sleeping time over this week is not influenced by the abnormal reduction in your weekdays’ sleeping time. Put differently, your preference is not changed through such a compensation process from the RPs (i.e., the weekdays) to the CP (i.e., the weekend). If this preference indifference can be identified by DMs, we say that the CP is the compensation independence of each RP. Given a life-long consumption ${\overrightarrow{c}}_{1,T}\in {\stackrel{\rightharpoonup }{C}}_{1,T}$, we are interested in the space $C_t\times C_k$ for $1\le s\le t<k\le T$. The primitive independence condition holds if $C_k$ is compensation independence of $C_t$. This condition is denoted as $C_t\propto C_k$ by using a directional symbol “$\propto$”, where the RP is in the left side and the CP in the right side.

Definition 1.

Primitive Compensation Independence.

We say that $C_k$ is compensation independence of $C_t$ if the preference invariance can be satisfied over the consumption space $C_t\times C_k$ for $1\le s\le t<k\le T$.

We are interested in the multiple RPs that correspond to the single CP in a compensating association. In our running example, the RP is multiple (e.g., Monday through Saturday) and the CP is singular (e.g., Sunday). If the multiple, prior dimensions are successive from s to t, our primitive condition can be represented as ${\overrightarrow{C}}_{s,t}\propto C_k$, which says that $C_k$ is compensation independence of a successive, prior dimension $C_i$ for $i=\left\{s,\dots ,t\right\}$. The following axiom can be directly derived under assuming the primitive condition over the space ${\overrightarrow{C}}_{s,t}\times C_k$.

Axiom PI.

Preference Invariance.

Given ${\overrightarrow{c}}_{1,T}\in {\stackrel{\rightharpoonup }{C}}_{1,T}$, the DM’s preference over ${\overrightarrow{c}}_{1,T}$ is invariant if ${\overrightarrow{C}}_{s,t}\propto C_k$ for $1\le s\le t<k\le T$.

The primitive compensation independence also implies that the compensation process $C_t\propto C_k$ is independent of the consumption levels other than $C_t$. The axiom can thus be formally represented over the space ${\overrightarrow{C}}_{s,t}\times C_k$ as below.

Axiom IIC.

Independence of Irrelevant Consumptions.

Given ${\overrightarrow{c}}_{1,T}\in {\stackrel{\rightharpoonup }{C}}_{1,T}$, the compensation process over $\propto$ is independent of the consumption levels expecting ${\overrightarrow{c}}_{s,t}$ if ${\overrightarrow{C}}_{s,t}\propto C_k$ for $1\le s\le t<k\le T$.

This axiom says that the compensation process over ${\overrightarrow{C}}_{s,t}\propto C_k$ is independent of the consumption levels in the spaces ${\overrightarrow{C}}_{1,s-1}$ and ${\overrightarrow{C}}_{t+1,T}$. Considering the present time $k$, ${\overrightarrow{C}}_{s,t}\propto C_k$ is irrelevant with: (i) the past consumptions ${\overrightarrow{c}}_{1,s-1}$ and ${\overrightarrow{c}}_{t+1,k-1}$; (ii) the present consumptions $C_k$; and (iii) the future consumptions ${\overrightarrow{c}}_{k+1,T}$. Similar axioms are also assumed in Rozen’s theory [5] (i.e., Axioms HC (iii), p. 1346) and HDB’s theory [24] (i.e., Axioms 9 and 12). Independence of present and future consumptions ensures that the CP is separable over time-nonseparable preferences. The present utility can be computed afresh as in time-separable preferences.

As before, we are interested in the space ${\overrightarrow{C}}_{s,t}\times C_k$. In this subspace of ${\stackrel{\rightharpoonup }{C}}_{1,T}$, $C_k$ is compensation independence of $C_i$ for $i=\left\{s,\dots ,t\right\}$. Meanwhile, the consumption level in $C_i$ has a change that triggers the compensation process from $C_i$ to $C_k$. After holding Axiom IID, Axiom PI equals to say that the DM’s preference over the sequence defined in ${\overrightarrow{C}}_{s,t}\times C_k$ is invariant in the process of compensation. Therefore, Axioms PI and IID together suggest that there exists a compensation process from ${\overrightarrow{C}}_{s,t}$ to $C_k$ once DMs identify ${\overrightarrow{C}}_{s,t}\propto C_k$, such that the preference invariance can be satisfied over the space ${\overrightarrow{C}}_{s,t}\times C_k$.

The axioms PI and IID directly derive from the condition of compensation independence under our behavioral assumption. However, they are insufficient for general models of satiation and habituation. We need to ask what kind of changes on the RP triggers the process of compensation. In other words, we need to particularize the form of compensation through proper conditions. Intuitively, if the change on RP is a reduction in the consumption levels, the end of compensation is to add an auxiliary quantity of consumption on CP.

Formally, the compensation is initiated by reductions in ${\overrightarrow{c}}_{s,t}$ to ${\overrightarrow{e}}_{s,t}$. There exists a particular sequence of consumption ${}_s{}^t\overrightarrow{\Delta }{}_k$ that can be compensated on $C_k$, such that preference invariance holds. The ${}_s{}^t\overrightarrow{\Delta }{}_k$ is called general compensatory consumptions. It can be written as ${\Delta }_k\left({\overrightarrow{c}}_{s,t},{\overrightarrow{e}}_{s,t}\right)$ to identify the consumption changes from ${\overrightarrow{c}}_{s,t}$ to ${\overrightarrow{e}}_{s,t}$ and directional associations from the RP dimensions ${\overrightarrow{C}}_{s,t}$ to the single CP dimension $C_k$. One can intuitively think that the reduction in RP influences DMs’ preference over the sequence, but this kind of effect is neutralized by compensating a ${}_s{}^t\overrightarrow{\Delta }{}_k$ on CP.

4. Satiation Compensation

This section introduces the main axiom in the satiation context. The hinge is to specify what kind of changes on RP that trigger the satiation compensation process. We will introduce the satiation compensation independence that is a particularization of the primitive independence condition. As a reminder of the clue in this paper, the different specifications on the change in the RP led to our hierarchy of preference conditions presented in the following section.

The satiation compensation process is triggered by an abstention on $C_i$, which says that the level of consumption $c_i$ reduces to 0, denoted as $c_i\to 0_i$. Definition 2 implies that there exists a quantity of consumption ${}_i{}^{}\phi {}_k\in C_k$ that can exactly compensate the prior abstention over $C_i$ such that preference invariance is held. The ${}_i{}^{}\phi {}_k$ is called a compensatory consumption of satiation concerning $C_i$. The preference condition $C_i{\propto }_{SCI}^{}{C}_k$ indicates that $C_k$ is Satiation Compensation Independence (SCI) of $C_i$.

Definition 2.

Satiation Compensation Independence.

The condition   $C_i{\propto }_{SCI}^{}{C}_k$   is held for   ${\overrightarrow{c}}_{1,T}\in {\stackrel{\rightharpoonup }{C}}_{1,T}$   where  $1\le i<k\le T$, if preference invariance on the consumption space is satisfied in the satiation compensation process before and after.

This preference condition can be represented as ${\overrightarrow{C}}_{s,t}{\propto }_{SCI}^{}{C}_k$ where $1\le s\le t<k\le T$, when the SCI condition is from the successively prior dimensions $C_i$ for $i=\left\{s,\dots ,t\right\}$ to the dimension $C_k$ as the common CP. Correspondingly, the compensatory consumption of satiation can be denoted as ${}_s{}^t\overrightarrow{\phi }{}_k\in C_k$ that corresponds to ${\overrightarrow{C}}_{s,t}$ as the RP. The SCI condition is a particularization of the primitive condition, where ${}_s{}^t\overrightarrow{\Delta }{}_k$ is particularized as ${}_s{}^t\overrightarrow{\phi }{}_k$. One can understand this satiation compensation process like a compensatory consumption on $C_k$ completely “assimilates” the corresponding reference stream ${\overrightarrow{c}}_{s,t}$. Under the assumption of cardinal preference over consumption sequences, this balance relation can be represented by the following axiom.

Axiom SC.

Satiation Compensation.

Considering ${\overrightarrow{C}}_{s,t}{\propto }_{SCI}^{}{C}_k$ for $1\le s\le t<k\le T$, there exists a compensatory consumption ${}_s{}^t\overrightarrow{\phi }{}_k\in C_k$, such that ${\overrightarrow{c}}_{s,t}~ \left({\overrightarrow{0}}_{s,t},{}_s{}^t\overrightarrow{\phi }{}_k\right)$, where ${}_s{}^t\overrightarrow{\phi }{}_k=S_k\left({\overrightarrow{c}}_{s,t},{\overrightarrow{0}}_{s,t}\right)$.

Axiom SC says that, if the condition ${\overrightarrow{C}}_{s,t}{\propto }_{SCI}^{}{C}_k$ is held, the reference stream ${\overrightarrow{c}}_{s,t}$ is preference indifferent with a particular consumption sequence defined in the space ${\overrightarrow{C}}_{s,t}\times C_k$ as $\left({\overrightarrow{0}}_{s,t},{}_s{}^t\overrightarrow{\phi }{}_k\right)$ that is called the compensation stream. The preference indifference is over two sequences that are of different lengths. The quantity ${}_s{}^t\overrightarrow{\phi }{}_k$ only depends on its reference stream ${\overrightarrow{c}}_{s,t}$. We define ${}_s{}^t\overrightarrow{\phi }{}_k=S_k\left({\overrightarrow{c}}_{s,t}\right)$ that emphasizes the changes of consumption levels from ${\overrightarrow{c}}_{s,t}$ to ${\overrightarrow{0}}_{s,t}$, where $S_k(.)$ is called a satiation function.

Axiom SC is the main axiom for a general SA model. By using Axiom SC together with Axiom PI and IIC, Proposition 1 preliminarily derives the period utility of the general SA model that is with additive structure.

Proposition 1.

The Additive Structure of Period Utility under SCI.

Considering   ${\overrightarrow{C}}_{s,t}{\propto }_{SCI}^{}{C}_k$   for   $1\le s\le t<k\le T$, if   $\succcurlyeq$   (preference relations) satisfies Axioms SC, PI, and IID, then The Period Utility can be represented by:

$${\mathcal{U}}_k\left(c_k\right)=u_k\left({}_s{}^t\overrightarrow{\phi }{}_k+c_k\right)-u_k\left({}_s{}^t\overrightarrow{\phi }{}_k\right)$$

where the satiation function ${}_s{}^t\overrightarrow{\phi }{}_k=S_k\left({\overrightarrow{c}}_{s,t}\right)$: ${\overrightarrow{C}}_{1,t}$→${ℝ}_{+}$ with $S_k\left({\overrightarrow{0}}_{s,t}\right)=0$ and $S_1=0$.

Proof. 

The proof is given in Supplementary Materials S1.1. □

The literature has widely assumed that $S_k(.)$ is linear, e.g., BS07, BS10, and HDB. Proposition 1 does not particularize the form of SF. We will derive the axiomatization for a linear SF in Section 7 and Section 8.

5. Habit-Based Satiation Compensation

This section presents the main axiom of satiation and habituation for an additive structure of period utility. Satiation compensation introduced in Section 4 is a key particularization of our primitive condition. In this section, we firstly introduce its counterpart—habit compensation—then combine satiation compensation and habit compensation to generate the presentation of the period utility under a hybrid preference condition.

Inspired by Gilboa and Schmeidler [21], a habitual consumption can be regarded as a neutral consumption that is formed from a prior consumption sequence. Considering ${\overrightarrow{c}}_{1,T}\in {\stackrel{\rightharpoonup }{C}}_{1,T}$, if a consumption $c_i\in C_i$ for $1\le i<T$ produces a neutral consumption on one of its subsequent dimensions $C_k$ for $i<k\le T$, we say that $C_k$ is Habit Compensation Independence (HCI) of $C_i$. Accordingly, the HCI condition can be represented as ${\overrightarrow{C}}_{s,t}{\propto }_{}^{HCI}{C}_k$ for $1\le s\le t<k\le T$, if $C_k$ is habit compensation independence of a successively prior dimension $C_i$ for $i=\left\{s,\dots ,t\right\}$, where we call ${\overrightarrow{C}}_{s,t}$ as the habit-based RP and $C_k$ as the habit-based CP.

The HCI condition is the other particularization of the primitive condition. It says that there exists a neutral (i.e., habitual) consumption on the CP if the consumptions on the RP are changed. Put differently, the HCI condition implies the existence of a habit compensation process that is triggered by the change on the RP. This process is ended by the existence of a neutral consumption of the marginal changes of the RP’s consumptions. This neutral consumption is called a habit-compensatory consumption, as denoted ${}_s{}^t\overrightarrow{\eta }{}_k\in ℝ$, which is correlated with the RP ${\overrightarrow{C}}_{s,t}$ and the CP $C_k$. Being consistent with Axiom SC, the marginal change is just ${\overrightarrow{c}}_{s,t}$ that is a reduction in ${\overrightarrow{c}}_{s,t}$ to ${\overrightarrow{0}}_{s,t}$. The ${}_s{}^t\overrightarrow{\eta }{}_k$ is a neutral consumption of ${\overrightarrow{c}}_{s,t}$ under the HCI condition. We present the axiom of habit compensation as follows.

Axiom HC.

Habit Compensation.

Considering ${\overrightarrow{C}}_{s,t}{\propto }_{}^{HCI}{C}_k$ for $1\le s\le t<k\le T$, there exists a compensatory consumption ${}_s{}^t\overrightarrow{\eta }{}_k\in C_k$, such that ${\overrightarrow{c}}_{s,t}~ \left({\overrightarrow{c}}_{s,t},{}_s{}^t\overrightarrow{\eta }{}_k\right)$, where ${}_s{}^t\overrightarrow{\eta }{}_k=h_k\left({\overrightarrow{c}}_{s,t}\right)$.

Considering ${\overrightarrow{C}}_{s,t}\times C_k$ in time-nonseparable preferences, the habituation effect leads to ${\overrightarrow{c}}_{s,t}\succcurlyeq \left({\overrightarrow{c}}_{s,t},0_k\right)$. Intuitively, a DM will refer ${\overrightarrow{c}}_{s,t}$ to $\left({\overrightarrow{c}}_{s,t},0_k\right)$ because she perceives the existence of an abstention on $C_k$. Axiom HC says that there exists a quantity ${}_s{}^t\overrightarrow{\eta }{}_k\in C_k$ that can rebalance DM’s preference such that holding ${\overrightarrow{c}}_{s,t}~ \left({\overrightarrow{c}}_{s,t},{}_s{}^t\overrightarrow{\eta }{}_k\right)$. Put differently, we can always find a ${}_s{}^t\overrightarrow{\eta }{}_k$ defined as the neutral consumption level that maintains DM’s preference invariance. Finding ${}_s{}^t\overrightarrow{\eta }{}_k$ on the CP $C_k$ based on the RPs ${\overrightarrow{c}}_{s,t}$ is known as a habit compensation process. The preference indifference is constructed on the different-length sequence $\left({\overrightarrow{c}}_{s,t},{}_s{}^t\overrightarrow{\eta }{}_k\right)$ that derives from the habit compensation over ${\overrightarrow{c}}_{s,t}$. It is easy to see that the quantity ${}_s{}^t\overrightarrow{\eta }{}_k$ solely depends on its reference stream ${\overrightarrow{c}}_{s,t}$. We define ${}_s{}^t\overrightarrow{\eta }{}_k=h_k\left({\overrightarrow{c}}_{s,t}\right)$, where $h_k(.)$ is called a habituation function.

Axiom HC is motivated by Gilboa and Schmeidler’s neutral continuation axiom [21], where they use it for a well-being model. It constructs preference indifference on different lengths of consumption sequences. This idea is also used in HDB’s axiomatization. Rozen’s structural axiom of habit formation [5] assumes that there exists a unique compensating stream such that DM’s choice behavior is invariant in whether there exists such a compensation stream as an endowment. This idea can be illustrated as two indifferent curves over the past consumptions, which is akin to Hicksian wealth compensation. Our structural Axiom HC does not work for Rozen’s theory because hers considers ordinal preferences over the same (i.e., infinite) length of sequences, Technically, Rozen [5] considers that two “habit” streams imposed by her Axiom HC have the same utility levels, whereas we consider that a habit stream and a habit-compensated stream that with a neutral consumption level have the same utility levels. Also, ours considers cardinal preference over the different (and finite) length of sequences. Wathieu [2] advocates that considering finite time horizons in models of habit formation could be more practical.

The literature in economics and management revealed a prevailing view: the effects of satiation and habituation are independent of each other. Put differently, past consumption could influence preference over current and future consumption via two distinct channels. Our behavioral assumption implies that satiation formation and habit formation can be interpreted by people’s psychology of compensation. The primitive condition is separated into the satiation compensation independence that leads to Axiom SC, and the habit compensation independence that leads to Axiom HC. The preference indifference constructed in two such axioms can be unified in ${\overrightarrow{C}}_{s,t}\times C_k$ where the common reference stream on ${\overrightarrow{C}}_{s,t}$ and the common corresponding compensation period $C_k$. The separated preference conditions can thus be combined as a new condition called habit-based satiation compensation independence (H-SCI). Denoted as ${\propto }_{SCI}^{HCI}$, it is a hybridization of the SCI condition ${\propto }_{SCI}^{}$ and the HCI condition ${\propto }_{}^{HCI}$. Satiation compensation under SCI says that the abstention of ${\overrightarrow{c}}_{s,t}$ can be balanced by the quantity ${}_s{}^t\overrightarrow{\phi }{}_k\in C_k$. Habit compensation under HCI says that the existence of ${\overrightarrow{c}}_{s,t}$ is obligated to compensate a quantity ${}_s{}^t\overrightarrow{\eta }{}_k\in C_k$. Two kinds of compensation have the same purpose: preference invariance.

What is the relationship between ${}_s{}^t\overrightarrow{\phi }{}_k$ and ${}_s{}^t\overrightarrow{\eta }{}_k$? The ${}_s{}^t\overrightarrow{\eta }{}_k$ is a neutral consumption of the marginal changes of consumption on the common RP. While a reduction over RP is triggered in the satiation formation context, the ${}_s{}^t\overrightarrow{\eta }{}_k$ is reduced on the common CP correspondingly. While the ${}_s{}^t\overrightarrow{\phi }{}_k$ assimilates ${\overrightarrow{c}}_{s,t}$ under SCI, it is obligated to assimilate concomitantly the ${}_s{}^t\overrightarrow{\eta }{}_k$ on $C_k$ under HCI, for preference invariance. One can understand that the auxiliary quantity ${}_s{}^t\overrightarrow{\phi }{}_k$ produced in satiation compensation process is based on a concomitant quantity $-{}_s{}^t\overrightarrow{\eta }{}_k$ produced in the habit compensation process. Consequently, the compositive compensatory consumption of satiation and habituation is given by ${}_s{}^t\overrightarrow{\Delta }{}_k={}_s{}^t\overrightarrow{\phi }{}_k-{}_s{}^t\overrightarrow{\eta }{}_k$ under ${\overrightarrow{C}}_{s,t}{\propto }_{SCI}^{HCI}{C}_k$. Since the satiation function ${}_s{}^t\overrightarrow{\phi }{}_k=S_k\left({\overrightarrow{c}}_{s,t}\right)$ and the habituation function ${}_s{}^t\overrightarrow{\eta }{}_k=h_k\left({\overrightarrow{c}}_{s,t}\right)$ defined before, we establish an adjustment function (AF) denoted as $S{h}_k(.)$, given by $S{h}_k\left({\overrightarrow{c}}_{s,t}\right)=S_k\left({\overrightarrow{c}}_{s,t}\right)-h_k\left({\overrightarrow{c}}_{s,t}\right)$ over the common ${\overrightarrow{c}}_{s,t}$ and $C_k$.

The satiation and habituation compensations are relatively independent of each other. Both have the identical target (i.e., DM’s preference invariance) and the same trigger that is the changes (reductions) on the RP’s consumptions. Concerning the common CP, two compensatory consumptions are inversely correlated. For modeling the net effect of satiation and habituation, under H-SCI, Proposition 1 in Section 4 can be extended to Proposition 2 as follows.

Proposition 2.

The Additive Structure of Period Utility under H-SCI.

Considering   ${\overrightarrow{C}}_{s,t}{\propto }_{SCI}^{HCI}{C}_k$   for   $1\le s\le t<k\le T$, if   $\succcurlyeq$   satisfies Axioms HC, SC, PI, and IID, then the Period Utility can be represented by:

$${\mathcal{U}}_k\left(c_k\right)=u_k\left({}_s{}^t\overrightarrow{\phi }{}_k-{}_s{}^t\overrightarrow{\eta }{}_k+c_k\right)-u_k\left({}_s{}^t\overrightarrow{\phi }{}_k\right)$$

  where the satiation function ${}_s{}^t\overrightarrow{\phi }{}_k=S_k\left({\overrightarrow{c}}_{s,t}\right)$: ${\overrightarrow{C}}_{1,t}$→${ℝ}_{+}$ with $S_k\left({\overrightarrow{0}}_{s,t}\right)=0$ and $S_1=0$; the habituation function ${}_s{}^t\overrightarrow{\eta }{}_k=h_k\left({\overrightarrow{c}}_{s,t}\right)$: ${\overrightarrow{C}}_{1,t}$→${ℝ}_{+}$ with $h_k\left({\overrightarrow{0}}_{s,t}\right)=0$ and $h_1=0$; and $\delta \in \left(0,1\right]$.

Proof. 

The proof is given in Supplementary Materials S2.1. □

The united functional formula can be given by ${\mathcal{U}}_k\left(c_k\right)=u_k\left({}_s{}^t\overrightarrow{\Delta }{}_k+c_k\right)-u_k\left({}_s{}^t\overrightarrow{\phi }{}_k\right)$, where the compositive compensatory consumption ${}_s{}^t\overrightarrow{\Delta }{}_k$ and the satiation-compensatory consumption ${}_s{}^t\overrightarrow{\phi }{}_k$. Proposition 1 is under SCI and thus particularizes ${}_s{}^t\overrightarrow{\Delta }{}_k={}_s{}^t\overrightarrow{\phi }{}_k$, while Proposition 2 is under H-SCI and thus particularizes ${}_s{}^t\overrightarrow{\Delta }{}_k={}_s{}^t\overrightarrow{\phi }{}_k-{}_s{}^t\overrightarrow{\eta }{}_k$. The ${}_s{}^t\overrightarrow{\Delta }{}_k$ can be interpreted as a reference-dependent satiation-compensatory consumption. The reference point is the habit-compensatory consumption ${}_s{}^t\overrightarrow{\eta }{}_k$. Proposition 2 degenerates Proposition 1 when ${}_s{}^t\overrightarrow{\eta }{}_k\to 0$. We thus call the compositive compensations the habit-based satiation compensation.

Habit compensation is independent of satiation compensation. The package of Axioms HC, PI, and IID has been a necessary and sufficient condition for a linear HF denoted as $h_k\left({\overrightarrow{c}}_{s,t}\right)={\displaystyle \sum }_{i=s}^t{\gamma }_{ik}{c}_i$ for some ${\gamma }_{ik}\in ℝ$ where $c_i$ is the marginal change in consumption on $C_i$. Vis-à-vis, the package of Axiom SC, PI, and IID cannot lead to a linear SF. By using the results of Section 4 and Section 5, Section 6 provides an axiomatic system of general SH models. Section 7 and Section 8 advance this system for further accommodating linear representations of satiation and habituation.

6. General Models of Satiation and Habituation

This section presents a necessary and sufficient condition for general SH models. The propositions in Section 4 and Section 5 are intermediate products. As a preliminary, we firstly emphasize the state space. Previous results consider the space ${\overrightarrow{C}}_{s,t}\times C_k$ for $1\le s\le t<k\le T$. By specifying $k=t+1$ in this section, the CP is the consecutively subsequent period of the RP as the space $\left\{C_s,\dots ,C_t,C_k\right\}$. For simplicity, this space is exhibited as $\left\{C_1,\dots ,C_{k-1},C_k\right\}$ by letting $s=1$ where the RP ${\overrightarrow{C}}_{1,k-1}$ and the CP $C_k$. The subspace ${\overrightarrow{C}}_{1,k}$ can be derived from ${\overrightarrow{C}}_{1,T}$ under the condition of $k=t+1$ for $1\le s\le t<k\le T$. Thus, the primitive condition can be simplified as ${\overrightarrow{C}}_{1,k-1}\propto C_k$.

If the compensation independence (CI) condition (i.e., SCI, HCI, and H-SCI) is held for any two consecutive spaces, this condition is called a successive CI condition. Readers can refer to our overall declaration of the state space in Section 2. The different settings of state spaces are for maintaining the generality in our axioms and the practicability in our theorems.

Motivated by the time impatience (TI) and the preference stationarity (PS), we first derive two additional axioms for general SA and SH models as general conditions in modeling the time preference. Axiom TI allows time discounting in the period utility when measuring the total utility. The literatures [30,31] and HDB assumed similar axioms. Axiom PS says that if two levels of consumption in two different periods are indifferent (e.g., $x_i$ and $y_{i-1}$ on $C_{i-1}\times C_i$), they should be perceived to be indifferent when the consumption in both periods is advanced by the same amount of time. HDB assumed a similar axiom.

Axiom TI.

Time Impatience.

Considering the consumption space   ${\overrightarrow{C}}_{1,k}$   , if   $x_i=x_{i+1}$   for any   $i\in \left\{1,\dots ,k-1\right\}$, then   $\left({\overrightarrow{0}}_{1,i}, x_{i+1}\right)\preccurlyeq \left({\overrightarrow{0}}_{1,i-1}, x_i\right)$ is valid.

Axiom PS.

Preference Stationarity.

Considering the consumption space ${\overrightarrow{C}}_{1,k}$, given $x_i\in C_i$ for any $i\in \left\{3,\dots ,k\right\}$, if $\left({\overrightarrow{0}}_{1,i-1}, x_i\right)~\left({\overrightarrow{0}}_{1,i-2}, y_{i-1}\right)$ is held for some $y_{i-1}\in C_{i-1}$, then $\left({\overrightarrow{0}}_{1,i-2}, x_{i-1}\right)~\left({\overrightarrow{0}}_{1,i-3}, y_{i-2}\right)$ when $x_i=x_{i-1}$ and $y_{i-1}=y_{i-2}$.

Axioms TI and PS indicate that the manner of aggregating the period utilities is a discounted sum. The indifference relations in satiation formation context like ${\overrightarrow{c}}_{s,t}~ \left({\overrightarrow{0}}_{s,t},{}_s{}^t\overrightarrow{\phi }{}_k\right)$ and in habit formation context like ${\overrightarrow{c}}_{s,t}~\left({\overrightarrow{c}}_{s,t},{}_s{}^t\overrightarrow{\eta }{}_k\right)$ are still held if advancing the identical time over the time horizon. HDB assumes similar axioms. The difference lies in that HDB assumes two similar packages of axioms that concern whether future consumptions are zero or not, but our theory assumes the axioms in a united way. Because the restriction—the last consumptions make no effect on the past compensation—has already been ensured by our primitive preference condition). Preference stationarity and time separability are two basic properties in the DU model. Under our independence conditions, time-nonseparable preference can be transferred as the time separability, and thus accepts related behavioral phenomena such as satiation and habit formation.

Hereinbefore, Propositions 1 and 2 are the necessary condition for Theorems 1 and 2, respectively. Axioms TI and PS are the common necessary condition for Theorem 1 and 2. These intermediate products together lead to the necessary and sufficient condition for general models of SA and SH. Note that the state space is restricted from the lifelong space ${\stackrel{\rightharpoonup }{C}}_{1,T}$ to the concerned space ${\overrightarrow{C}}_{1,k}$ by letting $k=t+1$ and rewriting $s=1$ for $1\le s\le t<k\le T$.

Theorem 1.

The General Satiation Model.

For   ${\overrightarrow{c}}_{1,k}\in {\overrightarrow{C}}_{1,k}$   where   $C_1{\overrightarrow{\propto }}_{SCI}^{}{C}_k$, the following statements are equivalent:

(A)

$\succcurlyeq$ satisfies Axioms SC, PS, TI, PI, and IID.

(B)

$\succcurlyeq$ can be represented by:

$$U\left({\overrightarrow{c}}_{1,k}\right)={\displaystyle \sum }_{i=1}^k{\delta }^{i-1}\left[u\left({}_1{}^{i-1}\overrightarrow{\phi }{}_i+c_i\right)-u\left({}_1{}^{i-1}\overrightarrow{\phi }{}_i\right)\right]$$

where SF: ${}_1{}^{i-1}\overrightarrow{\phi }{}_i=S_i\left({\overrightarrow{c}}_{1,i-1}\right)$ with $S_i\left({\overrightarrow{0}}_{1,i-1}\right)=0$, $S_1=0$, and $\delta \in (0,1]$.

Proof. 

The proof is given in Supplementary Materials S1.2. □

This axiomatic SA model contains several specific models of the literature as special cases. When the SF is always zero, our model degenerates into DU. When SF is specified as $S_i\left({\overrightarrow{c}}_{1,i-1}\right)=S_{i-1}\left({\overrightarrow{c}}_{1,i-2}\right)+c_{i-1}$, our model is specialized as Bell’s model [32]. When SF is specified as $S_i\left({\overrightarrow{c}}_{1,i-1}\right)=\xi \left(S_{i-1}\left({\overrightarrow{c}}_{1,i-2}\right)+c_{i-1}\right)$ for $\xi \in ℝ$, our model is specialized as BS07′s model. In Section 7, we will advance the current axiomatic system for a recursively defined linear SF that is a necessary and sufficient condition for BS07′s satiation model.

Theorem 2.

The General Satiation and Habituation Model.

For   ${\overrightarrow{c}}_{1,k}\in {\overrightarrow{C}}_{1,k}$   where   $C_1{\overrightarrow{\propto }}_{SCI}^{HCI}{C}_k$, the following statements are equivalent:

(A)

$\succcurlyeq$ satisfies Axioms HC, SC, PS, TI, PI, and IID.

(B)

$\succcurlyeq$ can be represented by:

$$U\left({\overrightarrow{c}}_{1,k}\right)={\displaystyle \sum }_{i=1}^k{\delta }^{i-1}\left[u\left({}_1{}^{i-1}\overrightarrow{\phi }{}_i-{}_1{}^{i-1}\overrightarrow{\eta }{}_i+c_i\right)-u\left({}_1{}^{i-1}\overrightarrow{\phi }{}_i\right)\right]$$

where SF: ${}_1{}^{i-1}\overrightarrow{\phi }{}_i=S_i\left({\overrightarrow{c}}_{1,i-1}\right)$ with $S_i\left({\overrightarrow{0}}_{1,i-1}\right)=0$, $S_1=0$, and HF: ${}_1{}^{i-1}\overrightarrow{\eta }{}_i=h_i\left({\overrightarrow{c}}_{1,i-1}\right)$ with $h_i\left({\overrightarrow{0}}_{1,i-1}\right)=0$, $h_1=0$; and $\delta \in \left(0,1\right]$.

Proof. 

The proof is given in Supplementary Materials S2.2. □

This axiomatic SH model can be degenerated into our axiomatic SA model when the HF is always zero. When the SF is always zero, our model degenerates into the general habituation (HA) model as given by $U\left({\overrightarrow{c}}_{1,k}\right)={\displaystyle \sum }_{i=1}^k{\delta }^{i-1}u\left(c_i-{}_1{}^{i-1}\overrightarrow{\eta }{}_i\right)$ for the HF: ${}_1{}^{i-1}\overrightarrow{\eta }{}_i=h_i\left({\overrightarrow{c}}_{1,i-1}\right)$ with $h_i\left({\overrightarrow{0}}_{1,i-1}\right)=0$, $h_1=0$, and $\delta \in \left(0,1\right]$. When the HF is specified as $h_i\left({\overrightarrow{c}}_{1,i-1}\right)=\gamma c_{i-1}+\left(1-\gamma \right)h_{i-1}\left({\overrightarrow{c}}_{1,i-2}\right)$, which is an exponential smoothing process of habit formation (e.g., [33]), the general HA model is specified as Wathieu’s habituation model [2,3]. Unlike Rozen’s habit compensation axiom [5] that corresponds to the infinite-and-identical-length sequences for ordinal preferences, our habit compensation axiom underlying the axiomatic HA model that corresponds to the finite-and-different-length sequences for cardinal preferences. Despite both Rozen’s and ours using the term “habit compensation” for the main axiom, there exist fundamental differences. Additionally, we allow the adjustment function $S{h}_i\left({\overrightarrow{c}}_{1,i-1}\right)=-h_i\left({\overrightarrow{c}}_{1,i-1}\right)$ to be negative, which departs from Rozen’s theory. Finally, our axiomatic SH model can be specialized as BS10′s model if SF is specified as $S_i\left({\overrightarrow{c}}_{1,i-1}\right)=\xi \left(S{h}_{i-1}\left({\overrightarrow{c}}_{1,i-2}\right)+c_{i-1}\right)$ and HF is specified as $h_i\left({\overrightarrow{c}}_{1,i-1}\right)={\displaystyle \sum }_{a=1}^{i-1}{\gamma }_{ai}{c}_a$, for $\xi ,{\gamma }_{ai}\in ℝ$. We will advance the current axiomatic system for linear representations of SF and HF in Section 8, which can be the necessary and sufficient condition for BS10′s SH model.

7. The Satiation Model with Linear Satiation Function

This section introduces a stronger version of Axiom SC (in Section 4), to establish the necessary and sufficient condition for the SA model with linear representations of satiation. Theorem 3 in this section is an advanced version of Theorem 1 (in Section 6) by imposing a recursively defined linear SF.

The strategy is to specify the compensation process assumed in our primitive condition. The accumulative satiation compensation process is triggered by a reduction on $C_i$ (i.e., $c_i\to e_i$) where $e_i$ is called a base consumption on $C_i$ for $c_i>e_i\ge 0_i$. Definition 3 implies that there exists a quantity of consumption ${}_i{}^{}\phi {}_k\in C_k$ that can exactly compensate the reduction over prior consumptions on $C_i$ for preference invariance. Similar to the SCI condition, the quantity ${}_i{}^{}\phi {}_k$ can be called a compensatory consumption of accumulative satiation. The preference condition $C_i{\propto }_{ASCI}^{}{C}_k$ indicates that $C_k$ is Accumulative Satiation Compensation Independence (ASCI) of $C_i$.

Definition 3.

Accumulative Satiation Compensation Independence.

The condition   $C_i{\propto }_{ASCI}^{}{C}_k$   is held for   ${\overrightarrow{c}}_{1,T}\in {\stackrel{\rightharpoonup }{C}}_{1,T}$   where $1\le i<k\le T$, if preference invariance on the consumption space is satisfied in the accumulative satiation compensation process before and after.

As before, we are interested in ${\overrightarrow{C}}_{s,t}{\propto }_{ASCI}^{}{C}_k$ for $1\le s\le t<k\le T$, where the RP $C_i$ for $i=\left\{s,\dots ,t\right\}$ and the common CP $C_k$. Such compensatory consumption can be denoted as ${}_s{}^t\overrightarrow{\phi }{}_k\in C_k$ that corresponds to ${\overrightarrow{C}}_{s,t}$ as the RPs. The base consumption of ${\overrightarrow{C}}_{s,t}$ can be denoted as ${\overrightarrow{e}}_{s,t}$ where ${\overrightarrow{c}}_{s,t}>{\overrightarrow{e}}_{s,t}\ge {\overrightarrow{0}}_{s,t}$. Compared with SCI, the ASCI condition imposes that the general CI condition can be valid for the more generalized process of compensation that is triggered by a reduction rather than an abstention. Therefore, ASCI is a stronger preference condition despite that both are particularizations of our primitive CI condition. One can consider that such a reduction is an abstention from the marginal reduction ${\overrightarrow{c}}_{s,t}-{\overrightarrow{e}}_{s,t}$ to the zero consumptions ${\overrightarrow{0}}_{s,t}$ over the RP ${\overrightarrow{C}}_{s,t}$. According to Definition-3, we propose Axiom ASC—a stronger version of Axiom SC.

Axiom ASC.

Accumulative Satiation Compensation.

Considering ${\overrightarrow{C}}_{s,t}{\propto }_{ASCI}^{}{C}_k$ for $1\le s\le t<k\le T$, there exists a compensatory consumption ${}_s{}^t\overrightarrow{\phi }{}_k\in C_k$, such that $\left({\overrightarrow{c}}_{s,t}-{\overrightarrow{e}}_{s,t}\right)~ \left({\overrightarrow{0}}_{s,t},{}_s{}^t\overrightarrow{\phi }{}_k\right)$, where ${}_s{}^t\overrightarrow{\phi }{}_k=S_k\left({\overrightarrow{c}}_{s,t},{\overrightarrow{e}}_{s,t}\right)$ for ${\overrightarrow{c}}_{s,t}>{\overrightarrow{e}}_{s,t}\ge {\overrightarrow{0}}_{s,t}\in {\overrightarrow{C}}_{s,t}$.

This axiom says that the marginal change defined as ${\overrightarrow{x}}_{s,t}={\overrightarrow{c}}_{s,t}-{\overrightarrow{e}}_{s,t}$ can be balanced by adding the quantity ${}_s{}^t\overrightarrow{\phi }{}_k$ on $C_k$. The SF is represented as ${}_s{}^t\overrightarrow{\phi }{}_k=S_k\left({\overrightarrow{c}}_{s,t},{\overrightarrow{e}}_{s,t}\right)$. When ${\overrightarrow{e}}_{s,t}={\overrightarrow{0}}_{s,t}$, it degenerates into ${}_s{}^t\overrightarrow{\phi }{}_k=S_k\left({\overrightarrow{c}}_{s,t}\right)$ of Axiom SC. The upper bound on ${\overrightarrow{e}}_{s,t}$ rules out the situation of ${\overrightarrow{c}}_{s,t}={\overrightarrow{e}}_{s,t}$, which says that the original consumption levels on the RPs must be changed (i.e., reduced). Supposing there exists $C_i$ for $i=\left\{s,\dots ,t\right\}$ such that $e_i=c_i$, the dimension $C_i$ is deemed as an “irrelevant past dimensions (consumptions)”. Rozen’s [5] Axiom HC(iii) and HDB’s Axiom 9 and 12 are also assumed for ruling out these “unchanged” outcomes. The spirit of these axioms is that preference over consumption sequences is independent of the identical consumption levels over the common period. Different from these predecessors, our axiomatization does not need such an additional axiom because Axiom IID and Axiom ASC have ensured the condition ${\overrightarrow{c}}_{s,t}>{\overrightarrow{e}}_{s,t}$.

Axiom ASC reveals that the satiation compensation process can only be triggered by the marginal changes of consumptions on the RPs. This process is independent of: (a) the dimensions that violate the CI condition; (b) the past dimensions in which consumption levels are not changed; (c) the present and future consumptions; and (d) the base consumptions after changes.

Example 1.

We use an example to illustrate the spirit of Axiom ASC. Suppose routinely you have 8 h of sleep per day that would make you feel comfortable. Taking one week as a time window, we have the sequence (8, 8, 8, 8, 8, 8, 8). At Friday night, you would feel tired since a lack of sleeping (because of a heavy workload), for example, saying the sequence from Monday to Thursday as (6, 8, 4, 3). Our interest is to know the consequence of a compensation process—how many hours for sleeping tonight that could mentally compensate for your missed time for sleeping before.

Such compensation is considered mentally or psychologically, rather than physically. You may feel that Monday is far too early for such compensation over Friday, then only Tuesday through Thursday are counted in. If so, you have exhibited a primitive CI condition ${\overrightarrow{C}}_{Wed, Thu}\propto C_{Fri}$. $C_{Mon}$ is thus ruled out because the compensating association between $C_{Mon}$ and $C_{Fri}$ cannot be established. In addition, $C_{Tue}$ is ruled out either because of the independence of irrelative past consumption regulated by Axiom IID.

The condition $\propto$ can be further particularized in Section 4 as ${\propto }_{SCI}^{}$, in Section 5 as ${\propto }_{}^{HCI}$, and in this section ${\propto }_{ASCI}^{}$. Considering the condition ${\overrightarrow{C}}_{Wed, Thu}{\propto }_{ASCI}^{}{C}_{Fri}$, we hold the base consumption $\left(4, 3\right)=C_{Wed}\times C_{Thu}$, the reference stream $\left(4, 5\right)=C_{Wed}\times C_{Thu}$, and the CP $C_{Fri}$. Additionally, we hold the SF: $\rho =S_{Fri}\left(\left(8, 8\right), \left(4, 3\right)\right)$ for the indifference $\left(4, 5\right)~\left(0, 0,\rho \right)$ by using Axiom ASC and $\left(8, 8, 8\right)~\left(4, 3,\rho +8\right)$ by using Axiom PI. Differently, the SF of Axiom SC is $\rho =S_{Fri}\left(\left(4, 5\right), \left(0, 0\right)\right)$. In the following, we will show that (1) like Proposition 3, the functional form of $S_{Fri}(.)$ is linear for ${\overrightarrow{C}}_{Wed, Thu}{\propto }_{ASCI}^{}{C}_{Fri}$, and (2) like Proposition 4, the linear $S_{Fri}(.)$ can be recursively defined for the condition $C_{Wed}{\overrightarrow{\propto }}_{ASCI}^{}{C}_{Fri}$ (i.e., $C_{Wed}{\propto }_{ASCI}^{}{C}_{Thu}{\propto }_{ASCI}^{}{C}_{Fri}$). Note that we use the space ${\overrightarrow{C}}_{1,k}≔{\times }_{i=1}^k{C}_i$ that equals to ${\overrightarrow{C}}_{s,t}\times C_k$ for $\left\{s,\dots ,t,k\right\}$. The successive independence condition is represented as $C_1\overrightarrow{\propto }{C}_k$. Proposition 3 derives a sufficient (and necessary) condition for the general linear representation of SF.

Proposition 3.

The General Linear Representation of SF.

Considering ${\overrightarrow{C}}_{1,k-1}{\propto }_{ASCI}^{}{C}_k$ for ${\overrightarrow{x}}_{1,k}\in {\overrightarrow{C}}_{1,k}$ given ${\overrightarrow{x}}_{1,k}={\overrightarrow{c}}_{1,k}-{\overrightarrow{e}}_{1,k}$ for ${\overrightarrow{c}}_{1,k},{\overrightarrow{e}}_{1,k}\in {\overrightarrow{C}}_{1,k}$, if $\succcurlyeq$ satisfies Axioms ASC, PI, and IID, then the satiation function can be represented by: $S_k\left({\overrightarrow{x}}_{1,k-1}\right)={\displaystyle \sum }_{i=1}^{k-1}{\xi }_{ik}{x}_i$ for some ${\xi }_{ik}\in ℝ$.

Proof. 

Supplementary Materials S3.1 provides the proofs with Lemmas S2–S4. □

This proposition implies three properties of a linear SF. First, the compensation from the RPs to the CP is additively separable (Lemmas S1 and S2): ${}_1{}^{k-1}\overrightarrow{\phi }{}_k={\displaystyle \sum }_{i=1}^{k-1}{}_i{}^{}\phi {}_k$. Second, the compensation is additively accumulable for each RP $C_i$ (Lemma S3):${\phi }_k\left(c_i, e_i\right)={\phi }_k\left(c_i, a_i\right)+{\phi }_k\left(a_i, e_i\right)$ where $c_i\ge a_i\ge e_i\ge 0_i$. Third, the compensation process is independent of the base consumption of each RP (Lemma S4): ${\phi }_k\left(x_i+e_i, x_i\right)={\phi }_k\left(e_i, 0_i\right)$ for any $x_i,e_i\in C_i$. Put differently, the compensation process is independent of the original profile of consumption in ${\overrightarrow{c}}_{1,k}$ and the base consumption ${\overrightarrow{e}}_{1,k-1}$, but only depends on the marginal changes of the reference stream ${\overrightarrow{x}}_{1,k-1}$. If substituting ASCI by using the successive ASCI condition, the linear SF shown in Proposition 3 can be recursively defined as shown in Proposition 4.

Proposition 4.

The Recursively Defined Linear Representation of SF.

Considering   $C_1{\overrightarrow{\propto }}_{ASCI}^{}{C}_k$   for   ${\overrightarrow{x}}_{1,k}\in {\overrightarrow{C}}_{1,k}$   given   ${\overrightarrow{x}}_{1,k}={\overrightarrow{c}}_{1,k}-{\overrightarrow{e}}_{1,k}$   for   ${\overrightarrow{c}}_{1,k},{\overrightarrow{e}}_{1,k}\in {\overrightarrow{C}}_{1,k}$, if   $\succcurlyeq$ satisfies Axioms ASC, PI, and IID, then the satiation function can be recursively defined as:

$$S_i\left({\overrightarrow{x}}_{1,i-1}\right)={\widehat{\xi }}_i\left(S_{i-1}\left({\overrightarrow{x}}_{1,i-2}\right)+x_{i-1}\right)$$

where ${\widehat{\xi }}_i\in ℝ$ for $i\in \left\{3,\dots ,k\right\}$, with $S_2\left(c_1\right)$ and $S_1$ for $i\in \left\{1,2\right\}$.

Proof. 

The proof is given in Supplementary Materials S3.2. □

Note that the package of Axiom ASC, PI, and IID in Propositions 3 and 4 is a sufficient (and necessary) condition for the additive structure of the period utility under ASCI as given by ${\mathcal{U}}_i\left(x_i\right)=u_i\left({}_1{}^{i-1}\overrightarrow{\phi }{}_i+x_i\right)-u_i\left({}_1{}^{i-1}\overrightarrow{\phi }{}_i\right)$ for $i\in \left\{3,\dots ,k\right\}$. The proof is akin to that of Proposition 1 in Section 4. Our interest here is in emphasizing the linear representation of ${}_1{}^{i-1}\overrightarrow{\phi }{}_i$. Propositions 3 and 4 include Axiom ASC as the main axiom. The only difference lies in whether the consumption space is “successive”.

We substitute the “stronger” successive ASCI condition for the successive SCI condition of Theorem 1. The axiomatic SA model with a recursively defined linear SF is exhibited in Theorem 1 as follows.

Theorem 3.

The SA Model with recursively defined linear SF.

For ${\overrightarrow{x}}_{1,k}\in {\overrightarrow{C}}_{1,k}$ where $C_1{\overrightarrow{\propto }}_{ASCI}^{}{C}_k$, the following statements are equivalent:

(A)

$\succcurlyeq$ satisfies Axioms ASC, PS, TI, PI, and IID.

(B)

$\succcurlyeq$ can be represented by:

$$U\left({\overrightarrow{x}}_{1,k}\right)={\displaystyle \sum }_{i=1}^k{\delta }^{i-1}\left[u\left({}_1{}^{i-1}\overrightarrow{\phi }{}_i+x_i\right)-u\left({}_1{}^{i-1}\overrightarrow{\phi }{}_i\right)\right]$$

where: ${}_1{}^{i-1}\overrightarrow{\phi }{}_i=S_i\left({\overrightarrow{x}}_{1,i-1}\right)={\widehat{\xi }}_i\left(S_{i-1}\left({\overrightarrow{x}}_{1,i-2}\right)+x_{i-1}\right)$ for ${\widehat{\xi }}_i\in ℝ$ and $i\in \left\{3,\dots ,k\right\}$, with $S_i\left({\overrightarrow{0}}_{1,i-1}\right)=0$, $S_1=0$, and $\delta \in \left(0,1\right]$.

Proof. 

The Proof is given in Supplementary Materials S3.3 by using Theorem 1 and Proposition 4. □

Note that the parameter ${\widehat{\xi }}_i$ in this Theorem can be understood as ${\xi }_{\left(i-1\right)i}$ over the space $C_{i-1}\times C_i$ when ${\overrightarrow{x}}_{1,i-2}={\overrightarrow{0}}_{1,i-2}$. If allowing a constant ${\widehat{\xi }}_i$, saying ${\widehat{\xi }}_i=\xi$ for any $i$, our axiomatic SA model can degenerate into BS07′s model.

8. The Hybrid SH Model with Linear SH Functions

This section introduces a necessary and sufficient condition for the hybrid SH model that is with linear representations of satiation and habituation. Theorem 4 is an advanced version of Theorem 2. The key is to consider the net effect of satiation and habituation. Axiom ASC is the main axiom in the context of satiation formation. However, in the habit formation context, we reformat Axiom HC into Axiom AHC. Following Axiom HC, the marginal changes in the reference dimensions produce a synchronously changed quantity of consumption over the compensation point. Following Axiom AHC, however, we consider the existence of the base consumption as follows.

Axiom AHC.

Accumulative Habit Compensation.

Considering ${\overrightarrow{C}}_{s,t}{\propto }_{}^{HCI}{C}_k$ for $1\le s\le t<k\le T$, there exists a compensatory consumption ${}_s{}^t\overrightarrow{\eta }{}_k\in C_k$, such that $\left({\overrightarrow{c}}_{s,t}-{\overrightarrow{e}}_{s,t}\right)~ \left(\left({\overrightarrow{c}}_{s,t}-{\overrightarrow{e}}_{s,t}\right),{}_s{}^t\overrightarrow{\eta }{}_k\right)$, where ${}_s{}^t\overrightarrow{\eta }{}_k=h_k\left({\overrightarrow{c}}_{s,t},{\overrightarrow{e}}_{s,t}\right)$ for ${\overrightarrow{c}}_{s,t}>{\overrightarrow{e}}_{s,t}\ge {\overrightarrow{0}}_{s,t}\in {\overrightarrow{C}}_{s,t}$.

There is no essential difference in the underlying preference conditions of Axiom AHC and Axiom HC. Both correspond to the reference stream ${\overrightarrow{c}}_{s,t}-{\overrightarrow{e}}_{s,t}$. If ${\overrightarrow{e}}_{s,t}={\overrightarrow{0}}_{s,t}$, Axiom AHC degenerates into Axiom HC. We distinguish them by calling an “accumulative habit compensation” over Axiom AHC yet retain the symbol “${\propto }_{}^{HCI}$”. The integrated condition of HCI and ASCI is called the habit-based ASCI, shorted by H-ASCI, and denoted as “${\propto }_{ASCI}^{HCI}$”. As illustrated in Section 5, the package of Axioms AHC, PI, and IID has already been a necessary and sufficient condition for a linear HF given by $h_k\left({\overrightarrow{c}}_{s,t},{\overrightarrow{e}}_{s,t}\right)={\displaystyle \sum }_{i=s}^t{\gamma }_{ik}\left(c_i-e_i\right)$ for some ${\gamma }_{ik}\in ℝ$.

For simplicity, we use the space ${\overrightarrow{C}}_{1,k}≔{\times }_{i=1}^k{C}_i$ that equals to ${\overrightarrow{C}}_{s,t}\times C_k$ for $\left\{s,\dots ,t,k\right\}$. The successive independence condition is given as $C_1\overrightarrow{\propto }{C}_k$. The sufficient (and necessary) condition for the general linear representation of adjustment function $S{h}_k(.)$ is provided by Proposition 5 as follows.

Proposition 5.

The General Linear Representation of Adjustment Function.

Considering ${\overrightarrow{C}}_{1,k-1}{\propto }_{ASCI}^{HCI}{C}_k$ for ${\overrightarrow{x}}_{1,k}\in {\overrightarrow{C}}_{1,k}$ given ${\overrightarrow{x}}_{1,k}={\overrightarrow{c}}_{1,k}-{\overrightarrow{e}}_{1,k}$ for ${\overrightarrow{c}}_{1,k},{\overrightarrow{e}}_{1,k}\in {\overrightarrow{C}}_{1,k}$, if $\succcurlyeq$ satisfies Axioms AHC, ASC, PI, and IID, then the adjustment function can be represented by: $S{h}_k\left({\overrightarrow{x}}_{1,k-1}\right)={\displaystyle \sum }_{i=1}^{k-1}{\omega }_{ik}{x}_i$ for some ${\omega }_{ik}\in ℝ$.

Proof. 

Supplementary Materials S4.1 provides the proofs with Lemmas S5–S7. □

Under $C_1{\stackrel{\rightharpoonup }{\propto }}_{ASCI}^{HCI}{C}_k$, the package of Axioms in Proposition 5 can derive: (1) the linear AF: $S{h}_i\left({\overrightarrow{x}}_{1,i-1}\right)={\displaystyle \sum }_{a=1}^{i-1}{\omega }_{ai}{x}_a$; and (2) the linear SF: $S_i\left({\overrightarrow{x}}_{1,i-1}\right)={\displaystyle \sum }_{a=1}^{i-1}{\lambda }_{ai}{x}_a$; and (3) the linear HF: $h_i\left({\overrightarrow{x}}_{1,i-1}\right)={\displaystyle \sum }_{a=1}^{i-1}{\gamma }_{ai}{x}_a$, for ${\omega }_{ai},{\lambda }_{ai},{\gamma }_{ai}\in ℝ$ and $i\in \left\{1,\dots ,k-1\right\}$. We hold $S{h}_i\left(x_a\right)=S_i\left(x_a\right)-h_i\left(x_a\right)$ and ${\omega }_{ai}={\lambda }_{ai}-{\gamma }_{ai}$ under $C_a{\propto }_{ASCI}^{HCI}{C}_i$. The linear SF can be recursively defined as shown in Proposition 6.

Proposition 6.

The Recursively Defined Linear Representation of SF.

Considering $C_1{\stackrel{\rightharpoonup }{\propto }}_{ASCI}^{HCI}{C}_k$ for ${\overrightarrow{x}}_{1,k}\in {\overrightarrow{C}}_{1,k}$, given ${\overrightarrow{x}}_{1,k}={\overrightarrow{c}}_{1,k}-{\overrightarrow{e}}_{1,k}$ for ${\overrightarrow{c}}_{1,k},{\overrightarrow{e}}_{1,k}\in {\overrightarrow{C}}_{1,k}$, if $\succcurlyeq$ satisfies Axioms AHC, ASC, PI, and IID, then the satiation function can be recursively defined, as:

$$S_i\left({\overrightarrow{x}}_{1,i-1}\right)={\widehat{\lambda }}_i \left(S{h}_{i-1}\left({\overrightarrow{x}}_{1,i-2}\right)+x_{i-1}\right)$$

where ${\widehat{\lambda }}_i\in ℝ$ for $i\in \left\{3,\dots ,k\right\}$, with $S_2\left(c_1\right)$ and $S_1$ for $i\in \left\{1,2\right\}$.

Proof. 

The proof is given in Supplementary Materials S4.2. □

Note that the package of Axiom AHC, ASC, PI, and IID in Propositions 5 and 6 is also a sufficient (and necessary) condition for the additive structure of the period utility under H-ASCI as given by ${\mathcal{U}}_i\left(x_i\right)=u_i\left({}_1{}^{i-1}\overrightarrow{\phi }{}_i-{}_1{}^{i-1}\overrightarrow{\eta }{}_i+x_i\right)-u_i\left({}_1{}^{i-1}\overrightarrow{\phi }{}_i\right)$ for $i\in \left\{3,\dots ,k\right\}$. The proof is akin to that of Proposition 2. Our interest here is in emphasizing the linear representation of ${}_1{}^{i-1}\overrightarrow{\phi }{}_i$ (and ${}_1{}^{i-1}\overrightarrow{\eta }{}_i$). Both Propositions 5 and 6 contain Axiom AHC and ASC as the main axioms. The only difference is whether the consumption space is “successive”. We substitute the “stronger” successive H-ASCI condition for the successive H-SCI condition. Theorem 4 exhibits the axiomatic SH model with a linear HF and a recursively defined linear SF.

Theorem 4.

The SH Model with Linear Representations of SF and HF.

For ${\overrightarrow{x}}_{1,k}\in {\overrightarrow{C}}_{1,k}$ where $C_1{\overrightarrow{\propto }}_{ASCI}^{HCI}{C}_k$, the following statements are equivalent:

(A)

$\succcurlyeq$ satisfies Axioms AHC, ASC, PS, TI, PI, and IID.

(B)

$\succcurlyeq$ can be represented by:

$$U\left({\overrightarrow{x}}_{1,k}\right)={\displaystyle \sum }_{i=1}^k{\delta }^{i-1}\left[u\left({}_1{}^{i-1}\overrightarrow{\phi }{}_i-{}_1{}^{i-1}\overrightarrow{\eta }{}_i+x_i\right)-u\left({}_1{}^{i-1}\overrightarrow{\phi }{}_i\right)\right]$$

where SF: ${}_1{}^{i-1}\overrightarrow{\phi }{}_i=S_i\left({\overrightarrow{x}}_{1,i-1}\right)={\widehat{\lambda }}_i \left(S{h}_{i-1}\left({\overrightarrow{x}}_{1,i-2}\right)+x_{i-1}\right)$, for ${\widehat{\lambda }}_i\in ℝ$ and $i\in \left\{3,\dots ,k\right\}$, with $S_i\left({\overrightarrow{0}}_{1,i-1}\right)=0$, $S_1=0$; and HF: ${}_1{}^{i-1}\overrightarrow{\eta }{}_i=h_i\left({\overrightarrow{x}}_{1,i-1}\right)={\displaystyle \sum }_{a=1}^{i-1}{\gamma }_{ai}{x}_a$ for some ${\gamma }_{ai}\in ℝ$ and $a\in \left\{1,\dots ,i-1\right\}$, with $h_i\left({\overrightarrow{0}}_{1,i-1}\right)=0$, $h_1=0$; and $\delta \in \left(0,1\right]$.

Proof. 

The Proof is given in Supplementary Materials S4.3 by using Theorem 2 and Proposition 6. □

When the HF is always zero, the representation of SF in Theorem 4 can be $S_i\left({\overrightarrow{x}}_{1,i-1}\right)={\widehat{\lambda }}_i\left(S_{i-1}\left({\overrightarrow{x}}_{1,i-2}\right)+x_{i-1}\right)$ that is identical to the recursively defined SF in Theorem 3. The general linear HF can be rewritten as $h_i\left({\overrightarrow{x}}_{1,i-1}\right)={\displaystyle \sum }_{a=1}^{i-1}\left({\lambda }_{ai}-{\omega }_{ai}\right)x_a$. The parameter ${\widehat{\lambda }}_i$ is reasonably understood as ${\lambda }_{\left(a=i-1\right)i}$ when ${\overrightarrow{x}}_{1,i-2}={\overrightarrow{0}}_{1,i-2}$. Suppose ${\widehat{\gamma }}_i={\gamma }_{\left(a=i-1\right)i}$ over $C_{i-1}\times C_i$ when ${\overrightarrow{x}}_{1,i-2}={\overrightarrow{0}}_{1,i-2}$ and given $S{h}_1=0$ and $S_1=0$. The general linear HF can be recursively defined as $h_i\left({\overrightarrow{x}}_{1,i-1}\right)={\widehat{\gamma }}_i{x}_{i-1}+\left(1-{\widehat{\gamma }}_i\right)h_{i-1}\left({\overrightarrow{x}}_{1,i-2}\right)$ that exhibits an exponential smoothing process of habit formation [33]. This special case of our axiomatic model is identical to [2,3] habituation model.

By further allowing a constant ${\widehat{\lambda }}_i$, say ${\widehat{\lambda }}_i=\lambda$ for any $i$, the recursively defined SF can be represented as $S_i\left({\overrightarrow{x}}_{1,i-1}\right)=\lambda \left(S{h}_{i-1}\left({\overrightarrow{x}}_{1,i-2}\right)+x_{i-1}\right)$. Our axiomatic model can degenerate into BS10′s satiation and habituation model. Additionally, we can ask whether there is any difference among ${}_1{}^{}\Delta {}_2$,${}_1{}^{}\Delta {}_3$, …,${}_1{}^{}\Delta {}_k$ under ${\overrightarrow{C}}_{1,k}$ where $C_1{\overrightarrow{\propto }}_{ASCI}^{HCI}{C}_k$. Concerning a fixed RPs, the compensatory consumption on different CPs—the time points of the future—is subject to the parameters ${\widehat{\lambda }}_i$ and ${\gamma }_{ai}$ of Theorem 4. Bounding $0\le {\widehat{\lambda }}_i,{\gamma }_{ai}\le 1$ has been assumed in the literature such as BS10.

9. Concluding Remarks

This paper has proposed an axiomatic foundation for satiation and habit formation. The main axioms are motivated by the notion of “compensation” that captures human’s psychological reaction. We flesh out that this behavioral assumption is intuitive and appropriate for modeling intertemporal preferences through a hierarchical preference condition. This study provides the most general theoretical framework of modeling satiation and habituation that has contained many existing utility models as special cases. The advanced axiomatic system further guarantees the linear representations of satiation, habituation, and their net effect (adjustment). This paper has contributed to modeling “nonstandard” time preference to the theoretical vein.

Throughout this paper, we assume a choice-based period utility by requiring non-negative consumptions. Our theory allows an experience-based period utility either. A typical scenario is to measure happiness over time stream of monetary or material payoffs [34,35,36]. The reference stream is a sequence of marginal changes of consumptions. For positive consumptions, all analyses of experience-based utility are identical to that of choice-based utility. Nevertheless, experience-based analyses allow negative outcomes in the sequence of consumptions or in the compensatory consumptions. One can consider that the independence condition turns out to be a kind of negative compensation, or in plain words, overdrawing the future account for the invariance of overall happiness. Note that any compensation process is always directional, from the past to the future. Therefore, compensation is always history-dependent.