André Lapied

Olivier Renault

Greqam Greqam

An Investigation of Time Consistency for Subjective Discounted Utility

Keywords: Discounting, Time Distance, Time Consistency, Separable Preferences JEL classification: D90

A well-known common agreement in decision theory is that only exponential decision makers are time consistent i.e. with the mere passage of time, future choices must not contradict the initial choice. Building on this result, a large range of works has studied time inconsistency as a direct application of hyperbolic discounting. These articles share the common objective time assumption under which decision makers have a perfect perception of future periods. This paper firstly highlights that, when no further condition than separability is mentioned, any discount mechanism is compatible with time consistency. Then, we investigate time consistency assuming that individual time perception may be submitted to time distortion. In particular, an axiomatic discounting model called Subjective Discounted Utility (SDU) is provided to illustrate how hyperbolic decision makers may be time consistent.

Introduction

Most of intertemporal decision models have focused on discounting, namely a psychological mechanism from which the decision maker translates her future instantaneous utility into a present instantaneous utility scale.

Discounting has huge economic implications as a whole: investment project, retirement strategy and environmental policy are salient examples that highlight the extensive ramifications of discounting to various fields of analysis. Economists have been firstly attracted by exponential discounting for its simplicity, since discount rates between two successive periods are assumed to be constant. Exponential discounting, associated with separability, is the central mainstay of the standard exponential discounted utility model [START_REF] Samuelson | A note on measurement of utility[END_REF], [START_REF] Koopmans | Stationary ordinal Utility and Impatience[END_REF]). But for a few decades, exponential discounting has been challenged by alternative behaviors called hyperbolic discounting. Empirical studies [START_REF] Thaler | Some Empirical Evidence on Dynamic Inconsistency[END_REF]) first revealed hyperbolic discounting whose key property is to catch decreasing impatience: hyperbolic agents exhibit steeper discounting for time intervals closer to the present. [START_REF] Frederick | Time discounting and time preference: A critical review[END_REF] pointed out the descriptive limits of exponential discounting and underlined the great diversity in elicited discount rate. Building upon these experimental evidences, theoretical works such as [START_REF] Loewenstein | Anomalies in intertemporal choice: Evidence and an interpretation[END_REF], [START_REF] Harvey | Proportional Discounting of future Costs and Benefits[END_REF], [START_REF] Laibson | Golden eggs and hyperbolic discounting[END_REF] 1 or [START_REF] Prelec | Decreasing impatience: A criterion for non-stationary time preference and 'hyperbolic' discounting[END_REF] have contributed to precisely define the meaning of decreasing impatience and hyperbolic discounting is now a standard concept in decision theory. Paradoxically, hyperbolic discounting did not succeed in replacing exponential discounting in most of theoretical models. The reason is directly related to the following question: if a decision maker (DM) has the possibility to revisit her previous choice, will she confirm the initial decision? If the answer is affirmative, then the DM is said to be time consistent in the sense that preferences reversal cannot occur with the mere passage of time. Conversely, if the answer is negative, the DM is said to be time inconsistent. Time inconsistency rises in situation of temptation or procrastination and traditional applications are undersaving [START_REF] Laibson | Golden eggs and hyperbolic discounting[END_REF]) or addiction [START_REF] Becker | A theory of rational addiction[END_REF]). [START_REF] Prelec | Decreasing impatience: A criterion for non-stationary time preference and 'hyperbolic' discounting[END_REF] stressed that time inconsistency induces inefficient choices due to unintended reversal of preferences for naïve agents, i.e. blinded individuals who do not anticipate the possibility to reconsider later their initial choice. That is why time consistency is traditionally viewed as a normative condition and as a firmly rooted component of intertemporal rationality. Since Strotz's seminal article (1956), it is well known that under a specific assumption called algebraic distance, a decision maker is time consistent if and only if she is exponential. Strotz's theorem has had a deep influence in intertemporal theory so that this conditional result (i.e. based on the algebraic distance) has progressively shifted toward an absolute "dogma" (i.e. dissociated from the algebraic distance) and the algebraic distance assumption has been progressively forgotten. This abusive common consensus has been designated by [START_REF] Drouhin | Hyperbolic discounting may be time consistent[END_REF] as a "conventional wisdom" and will be called in this paper the Intertemporal Folk Theorem (IFT). It is worth noticing that while radical changes have been introduced into intertemporal theory for a few decades -and especially the growing success of hyperbolic discounting -, the IFT has still remained a 1 Laibson has developed a quasi hyperbolic function where there is only decreasing impatience for the first period and for all other period the decision maker satisfies constant impatience. deeply rooted result, so that it is now a major cornerstone of intertemporal decision. The consensus conveyed by the IFT solves the previous paradox on discounting by imposing two restrictive interpretations. First the IFT confers to exponential discounting an absolute and exclusive relation with time consistency. Indeed, the IFT attributes to exponential behavior an unexpected normative agreement and, in spite of a poor descriptive power, exponential discounting keeps an unyielding comparative advantage on hyperbolic discounting. Second the IFT implies that hyperbolic discounting appears as the exclusive preferential cause to analyze misbehaviors related to time inconsistency. Given the great implications associated with Strotz's theorem, it is quite surprising that the algebraic distance gave rise to so slight interest in intertemporal decision. As a matter of fact, the algebraic distance is often substituted by a Stop Watch Time assumption equivalent to time preservation, namely, "the zero of time is resetting to the moment of decision". However we will show that the algebraic distance is a stronger assumption than Stop Watch time since the algebraic distance is equivalent to both objective time and time perception. Objective time is an implicit but decisive hypothesis shared by the great majority of intertemporal models [START_REF] Phelps | On Second-Best National Saving and Game Equilibrium Growth[END_REF], [START_REF] Harvey | The reasonableness of non-constant discounting[END_REF], [START_REF] Laibson | Golden eggs and hyperbolic discounting[END_REF] or [START_REF] Harris | Dynamic Choices of Hyperbolic Consumers[END_REF]). Objective time means that the DM has a perfect perception of future time intervals. For instance, the anticipated time duration of the coming month must be equal to the anticipated time duration for a month postponed in a decade. Nevertheless, as first noticed by A.C. Pigou, individuals often have "defective telescopic" ability to estimate far time intervals. In others words, individual's time perception may be submitted to time distortion and near time intervals may for example seem larger than equal more delayed time intervals. [START_REF] Albrecht | Hyperbolic Discounting Models in Prescriptive Theory of Intertemporal Choice[END_REF] introduced time perception into a discounted utility model. They assumed that exponential discounting is associated to objective time perception and then shown that deviation from standard discounting is fully explained by time perception distortion. In this way, hyperbolic agents have a logarithmic time perception. Zauberman and ali. (2009) claimed for a larger audience dedicated to time perception in intertemporal choices and [START_REF] Lapied | A Subjective Discounted Utility Model[END_REF] indicated that time perception is a mean to take into account for the great diversity in elicited discount rates. In this paper we plan to reconsider the Intertemporal Folk Theorem under the light of time perception. In particular, in reference to Strotz's seminal article, we introduce the concept of time distance as a dynamic version of time perception. Indeed, a time distance measures future time intervals depending on the observational decision period. Thus time distance both covers the definition of the time perception for the initial decision period, but also the transmission of time perception properties to the following decision period. For instance, the perception of future periods may be sensitive to the age of the DM, since time might seem running faster when the DM is getting older. The algebraic distance is then a special case of time distance for which no time distortion occurs. We build a model called Subjective Discounted Utility (SDU) in which the decision maker faces time in two distinct postures: time distance which associates to each objective future period a unique subjective future period and discounting that converses subjective future into present scale. By analogy to the Subjective Expected Utility (SEU) for risky choices, the Subjective Discounted Utility is a two-stage-decision model in which the DM discounts subjective periods. The main result associated to the SDU is that time consistency is reduced to the specific class of time distances that satisfies a triangular equality condition. This paper is organized as follows. Section 2 provides notations and a general representation with separability of time preferences. Section 3 will show that, without any additional condition, time consistency is consistent with all discount functions. The central interest of section 4 is to introduce time distances into discounted utility representation to reinterpret the time consistency condition and, building on these results, section 5 develops the Subjective Discounted Utility to propose a simple rule which characterizes time consistency.

Section 2 The Setting

Notations

A decision maker is first choosing between temporal prospects, i.e. sequences of future outcomes spread off over time. The income repartition between consumption and saving for the next year, the monthly allocation of time resource between labor and leisure and the organization for the next holiday week of pleasant and unpleasant tasks are salient examples of temporal prospects. With the mere passage of time, the DM has the possibility to reconsider later her initial plan. We assume a consequentialist naïve DM namely she has only a concern about future periods and does not anticipate the future opportunity to reconsider her first choice. The core interest of the paper is to determine the condition under which future choices do not contradict initial decision. The period set ࣪ is the set of non negative real numbers (with 0 as the initial period) and the DM's time horizon shall be ࣪ the set of non negative real numbers higher or equal to ݅ ∈ ࣪. For any decision period ݅ and given an index set ܵ = ሼ0,1,2, … , ݇, … ሽ, the time sequence set is ॻ = ሼ߬ ∶ ܵ → ࣪ with ∀݇ ∈ ܵ, ݅ ≤ ߬ < ߬ ାଵ ሽ and the outcome sequence set is ॿ = ሼܿ ∶ ܵ → ࣝሽ where ࣝ is a compact topological space 2 . The decision set for period ݅ is ∆ then ∆ = ॿ × ॻ where a choice alternative is a sequence of dated rewards as follows ሺ;ݔ ݐሻ = ሼሺܿ , ߬ ሻ, ሺܿ ଵ , ߬ ଵ ሻ, ሺܿ ଶ , ߬ ଶ ሻ, … ሽ. The decision set ∆ , endowed with the product topology, is connected and separable. A binary relation ≿ is defined on ∆ × ∆ for each decision period. As usual, ≻ and ∼ are defined to be respectively the asymmetric and symmetric parts of ≿ . For ݖ ∈ ࣝ, ሺܿ; ߬ሻ ∈ ∆ , the notation ሺܿ ;ݖ ߬ሻ = ሼሺܿ , ߬ ሻ, … , ሺܿ ିଵ , ߬ ିଵ ሻ, ሺ,ݖ ߬ ሻ, ሺܿ ାଵ , ߬ ାଵ ሻ … ሽ means that reward ܿ for period ߬ of the outcome sequence ሺܿ; ߬ሻ (with ߬ ∈ ߬) is replaced with the reward ݖ at the same period. Similarly the notation ൫ܿ , ,ݓ ;ݖ ߬൯ means that rewards ܿ and ܿ for periods ߬ and ߬ of the outcome sequence ሺܿ; ߬ሻ (with ߬ , ߬ ∈ ߬) are respectively replaced with rewards ݓ and ݖ at the same periods and so on. Lastly, a structural condition on time preference as a whole restricts our attention to the subset ∆ * which gathers all the choice alternatives ሺܿ; ߬ሻ verifying the following property: for any ߝ > 0, there exists an integer ܰ such that for all ݊ ≥ ܰ ሼሺܿ + ߝ, ߬ ሻ, ሺܿ ଵ , ߬ ଵ ሻ, … , ሺܿ , ߬ ሻ, ሺ0, ߬ ାଵ ሻ, ሺ0, ߬ ାଶ ሻ, … ሽ ≿ ሺܿ; ߬ሻ 2 As a consequence the set of possible outcomes is assumed the same for all periods.

ሺܿ; ߬ሻ ≿ ሼሺܿ -ߝ, ߬ ሻ, ሺܿ ଵ , ߬ ଵ ሻ, … , ሺܿ , ߬ ሻ, ሺ0, ߬ ାଵ ሻ, ሺ0, ߬ ାଶ ሻ, … ሽ

As stressed by [START_REF] Harvey | Proportional Discounting of future Costs and Benefits[END_REF], "the subset ሾ∆ * ሿ is the larger set on which an additive function converges and is a value function for the preference relation".

Representation of Time Preferences with Separability

We propose a representation theorem for time preferences mainly based on a separability axiom.

A1 (Regularity)

A regular binary relation on ∆ * is a continuous monotonic myopic weak order3 and has neutral element ܿ * in ॿ such that ∀߬, ߬′ ∈ ॻ and for all subset ܣ ⊆ ܵ, ሺܿ * ;ݖ ߬ሻ ∼ ሺܿ * ,ݖ ߬′ሻ (with ݖ ∶ ܣ → ࣝ).

A2 (Separability)

While regularity has to be seen as minimal condition for intertemporal rationality, separability is a strong assumption. Indeed, separability conveys the ideas that any dated reward can be estimated independently from each other (sequential independence) and that time estimation for dated reward is independent of what the reward is (magnitude independence).That is why separability axiom shall be divided into two distinct parts.

A2a (First Order Separability)

For all ,ݓ ,ݔ ,ݕ ݖ ∈ ࣝ, ܿ, ܿ′ ∈ ॿ , ߬ ∈ ॻ and ݅ ∈ ࣪ ൫ܿ ,ାଵ ,ݔ ;ݕ ߬൯ ≿ ൫ܿ ,ାଵ ,ݓ ;ݖ ߬൯ ⟹ ൫ܿ′ ,ାଵ ,ݔ ;ݕ ߬൯ ≿ ൫ܿ′ ,ାଵ ,ݓ ;ݖ ߬൯

First Order Separability (FOS) insures the absence of complementarities across successive periods4 . Associated with regularity, FOS implies that the order between outcome sequences ሺܿ * ௦ ;ݔ ߬ሻ can be entirely deduced from the order between dated rewards ሺ,ݔ ݏሻ. In the remaining analysis, we will use a slight abuse of notation for preferences defined on ࣝ × ࣪ writing ሺ,ݔ ݏሻ ≿ ሺ,ݕ ݐሻ and keeping in mind that Second Order Separability (SOS) is a traditional separability assumption as the Thomsen condition used by [START_REF] Fishburn | Time preference[END_REF]. However SOS says a little more since it implies that temporal aspects, i.e. both period ݐ and decision period ݅, of dated rewards may be isolated from outcome's estimation. We do not claim for a high descriptive degree associated with separability. Rather, assuming a very standard approach confers to our next results a stronger significance. We turn now to the third axiom retained for our representation theorem.

A3 (Impatience) The instantaneous utility function ݑ 5 is unique up to a positive linear transformation with ݑሺ0ሻ = 0.

Furthermore, decision weights satisfy ݉ ∶ ࣪ × ࣪ → ሿ0; 1ሿ, ݉ሺ݅; ݅ሻ = 1, ݉ሺ݅; ݐ ሻ > 0, ݉ is increasing with ݅ and decreasing with ݐ .

Section 3 Discounting and Time Consistency

Dynamic Discount Functions

For a few decades, literature dedicated to discounting has been producing a substantial flow of studies. Empirical studies reveal a sizeable diversity in elicited discount rates and the most frequent observed feature is decreasing impatience [START_REF] Frederick | Time discounting and time preference: A critical review[END_REF]) for hyperbolic agents. Theoretical works have integrated decreasing impatience in axiomatic models [START_REF] Loewenstein | Anomalies in intertemporal choice: Evidence and an interpretation[END_REF]) so that hyperbolic discounting is now very standard in intertemporal choices. Nevertheless, in spite of a poor descriptive power, exponential discounting (or constant impatience) paradoxically remains the privileged way for economic applications dealing with optimal behavior.

Definition 2 (Discount Functions)

A discount function ߤ ∶ ࣪ → ሿ0,1ሿ is a continuous function which simultaneously satisfies the three following properties:

(d1) ߤሺ0ሻ = 1 (Normalization) (d2) ߤሺݐ ሻ > 0 (Positive discounting) (d3) ߤ is (weakly) decreasing with ݐ (Impatience)
Property (d1) says that there is no discount for the present period and (d2) implies that the DM has minimal concern for all future periods, no matter how far from the present are. Lastly (d3) exhibits impatience.

Definition 3 (Discounted Utility)

We call Discounted Utility (DU) a separable intertemporal utility function whose decision weights are discount functions as presented in definition 2.

Observation 3.1

The intertemporal function ( 3) is a Discounted Utility (DU) since ݐ∀ ∈ ࣪ , decision weights are dynamic discount functions:

݉ሺ݅; ݐሻ = ߤ ሺݐ -݅ሻ ሺ4ሻ

5 Observe that ݑ is independent of time. The critical point is that the instantaneous utility must be independent from the decision period ݅. As noticed by [START_REF] Drouhin | Hyperbolic discounting may be time consistent[END_REF], instantaneous utility independence from period ߬ is not a necessary condition and dropping this assumption does not perturb our next results.

The great heterogeneity in elicited discount rates can be expressed by the following axiom. Each discount mechanism is uniquely associated to an discount premium ߠ.

A4 (Time Discounting)

,ݏ∀ ݐ ∈ ࣪, ∃ߠ = ߠሺ,ݐ ݇ሻ: ࣪ × ࣪ → ࣪ such that:

Time consistent choices

There is an old tradition in intertemporal choice for associating hyperbolic discounting to time inconsistency [START_REF] Thaler | Some Empirical Evidence on Dynamic Inconsistency[END_REF], [START_REF] Laibson | Golden eggs and hyperbolic discounting[END_REF]). Time inconsistency rises when the mere passage of time induces preferences reversal. For instance, [START_REF] Akerlof | Procrastination and Obedience[END_REF] studied the case of procrastination for which an initial "planner" chooses to accomplish an unpleasant task in remote future (say one month), but future "doer" reveals preference for delaying the unpleasant task (say one additional month). Therefore, future doer continually contradicts the initial plan and procrastination leads to the perpetual delaying of unpleasant tasks. Hyperbolic discounting is generally seen as a preferential source of procrastination since short term preference exhibits higher discount rate than long term preference. On the contrary, time consistency guaranties the absence of preferences reversal across decision periods. In other words, with the mere passage of time, the future "doer" does not contradict the choice of the initial "planner". [START_REF] Prelec | Decreasing impatience: A criterion for non-stationary time preference and 'hyperbolic' discounting[END_REF] noticed that time inconsistency induces inefficient choices due to unintended reversal of preferences for naïve agents. As a consequence, time consistency appears as a sound normative condition for rational choices. Proposition 3.2 merely shows that without any additional condition, time consistency is compatible with any discount mechanism. For instance, assume ߤ ሺݐሻ = ߣ ௧ ഀ ሺߙ > 0ሻ then corresponding time consistent decision weights will be ݉ሺ݅; ݐሻ = ߣ ௧ ഀ ି ഀ . Observe that the previous discount function covers the different cases for impatience6 because if ܽ < 1 (resp. ܽ > 1, ܽ = 1) then decreasing (resp. increasing, constant) impatience holds.

A5 (Time Consistency)

Corollary 3.3

Let ≿ verify time consistency. Then for decision weights defined as in theorem 2.1, the two following statements are equivalent: . Thus preferences reversal may hold since the DM both exhibits ൫ܿ * ଵ,ଶ -100,180; ߬൯ ≻ ሺܿ * ; ߬ሻ and ሺܿ * ; ߬′ሻ ≻ ଵ ൫ܿ * ଵ,ଶ -100,180; ߬′൯ with ߬′ = ߬ -ሼ߬ ሽ.

ሺ * ሻ ,ݏ∀ ݐ ∈ ࣪ , ݏ ≤ ,

Section 4 Time Distances

Our departure point for this section is an observation by [START_REF] Drouhin | Hyperbolic discounting may be time consistent[END_REF]. Consider the following decision weight ݉ሺ݅; ݐሻ = ሺ1 + ݅ሻ ሺ1 + ݐሻ ⁄ then by proposition 3.2, ݉ satisfies time consistency with ߤ ሺݐሻ = 1 ሺ1 + ݐሻ ⁄ and hyperbolic decision makers may be time consistent.

Like SWT for EDU, the objective of this section is to determine a specific condition under which HDU is the only preference pattern compatible with time consistency. More precisely, this section will provide a general condition defining the exclusive association between time consistency and a specific DU model.

The first economist dealing with time consistency for naïve agents was [START_REF] Strotz | Myopia and inconsistency in dynamic utility maximisation[END_REF]. The key element of his demonstration is the algebraic distance which imposes that discount weights only depends on the algebraic difference between discounted period and decision period (݉ሺ݅; ݐሻ = ߤሺݐ -݅ሻ). Strotz's famous theorem states that under algebraic distance, only exponential decision makers are time consistent. Undoubtedly, there is a close connection between Strotz's theorem and Observation 3.4 so that intertemporal theory traditionally considers the two assumptions as equivalent. However, as we will see, the algebraic distance is a stronger assumption and contains SWT. Such an abusive equivalence is called in this paper "Intertemporal Folk

Theorem". We argue that this abusive assimilation is due to the fact that like SWT, algebraic distance in Strotz's methodology is a pure formal assumption without any behavioral perspective. In order to clearly dissociate algebraic distance from SWT and to break with the Intertemporal Folk Theorem, we revive Strotz's seminal article by including the algebraic distance in a larger concept named time distance.

Time distance is a dynamic version of time perception. Time distance measures at decision period ݅ the gap between decision period ݅ and future period .ݐ Therefore if ߩሺݐሻ denotes the DM's time perception, then ݀ሺ݅; ݐሻ = ߩ ሺݐ -݅ሻ defines her time distance. Intertemporal theory allows a growing interest to time perception [START_REF] Albrecht | Hyperbolic Discounting Models in Prescriptive Theory of Intertemporal Choice[END_REF], Zauberman and ali. (2009)). The introduction of time perception into standard discounting fulfills the wish of giving a sound behavioral basis to recent changes observed in time discounting and of explaining the great diversity in elicited discount functions. In this way, time distances capture two behavioral schemes. The estimation of one year from now may have sensibly different treatment than one year in one decade (ߩሺ1ሻ ≠ ߩሺ11ሻ -ߩሺ10ሻ). In addition the young DM's consideration for one year in one decade may be sensibly different from the same consideration brought by older DM (ߩ ሺ11ሻ -ߩ ሺ10ሻ ≠ ߩ ሺ11ሻ -ߩ ሺ10ሻ ሺ݅ < ݆ሻ). The first example suggests that for a given observational time point, our perception of future duration may be sensitive to the distance from the observational time point and for instance the DM is more or less unable to mentally represent far future. The second example highlights that our perception of future time intervals may be sensitive to the observational time point and consideration for far future depend on the age of the DM.

In spite of the great heterogeneity allowed by time distortion, time distance functions share a limited set of properties.

Definition 5 (Time Distance Functions)

A time distance function ݀ ∶ ࣪ × ࣪ → ࣪ is a continuous function which simultaneously satisfies the three following properties:

(p1) ݀ሺ݅; ݐሻ = ߩ ሺݐ -݅ሻ ≥ 0 (Dynamic Time Perception) (p2) ݀ is decreasing with ݅ and increasing with ݐ (p3) ݀ሺ݅; ݅ሻ = 0 (Normalization for present period)

Property (p1) says that time distance is a subjective period and subjectivity may depend on decision period.

Property (p2) insures that time distance decreases (resp. increases) when the DM is moving closer to (resp.

away from) the discounted period. Normalization implies that the present is not submitted to time distortion.

Dynamic discount weights (ߤ ሺݐ -݅ሻ), as in Observation 3.1, cannot be easily deduced from static discount weights (ߤ ሺݐሻ) yet. Indeed, observe that decision weights are discount functions of period gap and not only of period. Thus, period gap may be dependent from the decision period. The next observation will clarify this point. For initial condition, we focus on convexity or concavity properties of time perception. Salient cases are

Objective Time (ߩሺݐሻ = )ݐ or Logarithmic Time Perception (ߩሺݐሻ = lnሺ1 + ݐሻ). For transmission condition, if ߮ሺ݅; ݐሻ = 0 then the time distance satisfies Time Preservation (ߩ ሺݐ -݅ሻ = ߩሺݐ -݅ሻ). For ߮ሺ݅; ݐሻ > 0, we will say that the DM exhibits Time Compression and if ߮ሺ݅; ݐሻ < 0 that the DM exhibits Time Extension.

Proposition 4.2

All time distance function is uniquely defined by a combination of an initial condition and a transmission condition.

For instance, the algebraic distance ݀ሺ݅; ݐሻ = ݐ -݅ and the quasi algebraic distance ݀ሺ݅; ݐሻ = ሺݐ -݅ሻ ሺ1 + ݅ሻ ⁄ share the Objective Time property but have different transmission conditions. The algebraic distance and the logarithmic time distance ݀ሺ݅; ݐሻ = lnሺ1 + ݐ -݅ሻ share the time preservation condition but reflect different time perceptions. Corollary 4.4 summarizes these observations. But in order to prove it, we need to introduce, as for discounting, a premium ߬ which will characterize all time distances. This is precisely the objective of the next axiom. The previous section highlighted the restrictive implication of SWT. Observation 3.4 is often assimilated to Strotz's theorem so that SWT and the algebraic distance are traditionally viewed as equivalent assumptions.

A6 (Gap

However this assimilation is abusive since the algebraic distance is a stronger assumption.

Observation 4.5

For separable intertemporal utility functions as presented in definition 1, Time preservation is equivalent to SWT.

Actually, SWT is only one part of the algebraic distance. We now turn out the main theorem of this section and explain why this abusive association is frequently observed in intertemporal decision theory.

Theorem 4.6

For each given discount mechanism ߤ , there is a set of time distances such that the decision maker is time consistent.

݀ሺ݅; ݐሻ = ߤ ିଵ ቆ ߤ ሺݐሻ ߤ ሺ݅ሻ ቇ ሺ9ሻ
The main theorem is the largest result for time consistency based on discounting and time distance. The counterpart of this generality is a relative indetermination since a single decision weight ݉ as in ( 6) can be obtained from different combinations of discount function and time distance function. For example, ݉ሺ݅; ݐሻ = ሺ1 + ݅ሻ ሺ1 + ݐሻ ⁄ may formally result from the combination between exponential form for ߤ and the following time distance ݀ሺ݅; ݐሻ = ݎ ିଵ ሾlnሺ1 + ݐሻ -lnሺ1 + ݅ሻሿ or from the combination between hyperbolic form for ߤ and quasi hyperbolic distance ݀ሺ݅; ݐሻ = ሺݐ -݅ሻ ሺ1 + ݅ሻ ⁄ . The reason for that indetermination is the absence of a behavioral correspondence between the two temporal postures. Indeed, observe that in this approach, the function ߤ may be independently chosen from the time distance. In the next section, the time distance fully determines the nature of discounting. Actually, the expression (9) associates to each discount mechanism ߤ a specific class of time distances such that the DM is time consistent. It is worth noticing that theorem 4.6 may be strengthened by imposing only one additional core property of the algebraic distance.

Corollary 4.7

If (TP) is retained for (9) then only exponential DM are time consistent ߤ ሺݐሻ = ߣ ௧

If (OT) is retained for (9) then ߤ ሺݐሻ = ߤሺݐሻ, and a DM is time consistent provided that ݀ሺ݅; ݐሻ = ߤ ିଵ ሺߤሺݐሻ ߤሺ݅ሻ ⁄ ሻ

First part of corollary shows that the Intertemporal Folk Theorem is a very restrictive version of a larger condition associated to time consistency. As a consequence, when time preservation (or SWT) is dropped, EDU loses the exclusive association with time consistency. Second part of corollary shows that, in a standard approach (OT), all decision makers can be time consistent and each discount mechanism is linked with a unique transmission condition. For instance, hyperbolic agents are time consistent if and only if they have a quasi algebraic distance. However, the second part of corollary 4.7 still remains too restrictive since the diversity of time perceptions is denied. In other words, Objective Time prevents us from building a sound behavioral structure for time consistent decision makers. The main theorem and its corollary suggest that a serious track to generalize Strotz's Theorem is relaxing the objective time assumption. To achieve this goal, the introduction of time perception into intertemporal models appears as an appealing way to consider subjective time.

Section 5 Subjective Discounted Utility

Theorem 4.6 is the largest result dealing with time consistency and based on time distance and discounting.

However, it is a pure formal result with poor behavioral interpretation. In this section, we present a model in which discounting is fully explained by the nature of time distance. In other words, decision weights can be expressed in the following sense: for

ߣ = ݁ ି ሺݎ > 0ሻ, ݐ∀ ∈ ࣪ , ݅ ∈ ࣪ ݉ሺ݅; ݐሻ = ߣ ௗሺ;௧ሻ ሺ10ሻ
The Subjective Discounted Utility (SDU) was introduced by Lapied and Renault (2012) for a unique decision period model. The main interest of such a model is that each discount function is associated to a unique time perception. In other words, the SDU approach confers to individual discounting a clear behavioral interpretation. For instance, a pioneer article by [START_REF] Albrecht | Hyperbolic Discounting Models in Prescriptive Theory of Intertemporal Choice[END_REF] demonstrates that, as the famous "law" of diminishing marginal utility, hyperbolic agents have a logarithmic time perception, i.e. the marginal perceived time is a decreasing function of time. The SDU is based on a subjective discounting axiom. We will show that this axiom is necessary to get the desired form (10) for decision weights defined as in Observation 3.1.

A7 (Subjective Discounting)

,ݏ∀ ݐ ∈ ࣪, ∃ߩ: ࣪ → ࣪ such that:

ሺ,ݔ ݏሻ ~ ሺ,ݕ ݐሻ ⟹ ቀ,ݔ ߩ ିଵ ൫ߩሺݏሻ + ߩሺ݇ሻ൯ቁ ~ ቀ,ݕ ߩ ିଵ ൫ߩሺݐሻ + ߩሺ݇ሻ൯ቁ
The SD axiom implies a special representation for decision weights at initial period ݉ሺ0; ݐሻ = ߣ ఘሺ௧ሻ [START_REF] Lapied | A Subjective Discounted Utility Model[END_REF]). Furthermore, if ݏ = 0 then A7 becomes ሺ,ݔ 0ሻ ~ ሺ,ݕ ݐሻ ⟹ ሺ,ݔ ݇ሻ ~ ቀ,ݕ ߩ ିଵ ൫ߩሺݐሻ + ߩሺ݇ሻ൯ቁ.

The meaning of A7 is quite easy to perceive. First, all discount axioms like A4 involve objective time in both members of the proposition. Conditions in definition 5 ensure that ߩ ିଵ does exist and if ߩ converts all real periods in subjective dates then ߩ ିଵ produces the reciprocal transformation namely each subjective date is associated to a unique objective period. Furthermore the expression ߩሺݐሻ + ߩሺ݇ሻ can be interpreted in a traditional way considering ߩሺ݇ሻ as an additional subjective delay. To illustrate, suppose that a 90 immediate reward is the same than obtain 100 in one period (say one year) so ሺ90,0ሻ ~ ሺ100,1ሻ. Then A7 implies for a two period decay that ሺ90,2ሻ ~ ቀ100, ߩ ିଵ ൫ߩሺ1ሻ + ߩሺ2ሻ൯ቁ. Of course, it is not necessary that ߩ ିଵ ൫ߩሺ1ሻ + ߩሺ2ሻ൯ = 3. If the case, then ߩሺݐሻ = ,ݐ i.e. time perception is reduced to objective time7 and A7 changes for

Corollary 5.2

In the SDU model, for all decision periods, the time distance integrally determines the nature of discounting for decision weights defined as in Observation 3.1

By theorem 5.1 and corollary 5.2, the SDU insures that there exists a one-to-one correspondence between time distance and dynamic discount weight. As a consequence, discount properties are fully explained by behavioral features conveyed by time distance functions. That is why the subjective discounted utility appears as a simple means to find out a general condition for time consistency exclusively based on time distance properties.

Looking back to the two previous time distances, what is the key property shared by both exponential and hyperbolic consistent agents?

We start from the Intertemporal Folk Theorem from which under the algebraic distance, only exponential decision makers can be time consistent. Intuitively, if one of the two conditions of the algebraic distance is weakenedsay for instance objective time -then we will show that time preservation must be weakened in a very precise way such that the DM remains time consistent. We first characterize time consistency in the SDU model by a simple rule based on time distance properties and figure 1 plotted below will illustrate the intimate relation between time perception (initial condition) and the transmission condition. Then, we will reinforce this central interpretation by developing the meaning of the expression "very precise way" with the help of time premiums ߠ and ߬ respectively presented in section 3 and section 4 of the study.

Recall that time distances are singular distance functions with respect to mathematical distances defined in the usual sense. In particular, the natural ordering introduced by time elapsing imposes that time distances are not symmetric. However, the singularity of time distance may also be highlighted by the use of the traditional triangular inequality property ݀ሺ݅; ݐሻ ≤ ݀ሺ݅; ݏሻ + ݀ሺ;ݏ ݐሻ shared by all "spatial" distances. Indeed, due to potential distortion, time distances permit that the reverse triangular inequality such as ݀ሺ݅; ݐሻ > ݀ሺ݅; ݏሻ + ݀ሺ;ݏ ݐሻ. For instance, the following the quasi algebraic distance ݀ሺ݅; ݐሻ = ሺݐ -݅ሻ ሺ1 + ݅ሻ ⁄ implies ݀ሺ݅; ݐሻ > ݀ሺ݅; ݏሻ + ݀ሺ;ݏ ݐሻ.

Definition 5 (Triangular Inequality for Time Distances)

For all ,ݏ ݐ ∈ ࣪ , ݅ ∈ ࣪, ݀ሺ݅; ݐሻ ቆ > = < ቇ ݀ሺ݅; ݏሻ + ݀ሺ;ݏ ݐሻ ሺ12ሻ
The traditional part of the triangular inequality (≤) may be interpreted as usual. For the reverse triangular inequality (>), the following intuitive interpretation holds. Consider an agent who has to return with delay, say two months, some report. The agent plans to work during the second month and considers that this time interval is sufficient to do the job. After one month elapsing, the coming month seems to be shorter so that the DM begins to doubt her ability to return the report in the right delay.

In most of the studies dealing with time consistency, objective time and time preservation are simultaneously retained and the Intertemporal Folk Theorem holds. It is easy to show that Objective Time implies ݀ሺ݅; ݐሻ -݀ሺ݅; ݏሻ = ݀ሺ݅; ݅ + ݐ -ݏሻ and that time preservation implies ݀ሺ;ݏ ݏ + ݇ሻ = ݀ሺ݅; ݅ + ݇ሻ(for ݀ well defined). By associating the two previous expressions, the triangular equality holds ݀ሺ݅; ݐሻ = ݀ሺ݅; ݏሻ + ݀ሺ;ݏ ݐሻ. However, Objective Time and Time Preservation have a poor descriptive basis and observations tend to confirm two alternative time postures. On the one hand, the observation of defective telescopic abilities of individuals tends to claim for logarithmic time perceptions. On the other hand, empirical studies frequently reveal time compression, i.e. there is a larger propensity for older agents to reduce anticipated coming time intervals than for younger agents. Hence, logarithmic time perception insures that ݀ሺ݅; ݐሻ -݀ሺ݅; ݏሻ < ݀ሺ݅; ݅ + ݐ -ݏሻ and time compression that ݀ሺ;ݏ ݏ + ݇ሻ < ݀ሺ݅; ݅ + ݇ሻ. Therefore if we consider the case for ݇ = ݐ -ݏ then ݀ሺ݅; ݐሻ -݀ሺ݅; ݏሻ < ݀ሺ݅; ݅ + ݇ሻ and ݀ሺ;ݏ ݐሻ < ݀ሺ݅; ݅ + ݇ሻ simultaneously result. Assume in addition that the previous inequalities reveal exactly the same intensity difference in the sense that ݀ሺ݅; ݐሻ -݀ሺ݅; ݏሻ = ݀ሺ;ݏ ݐሻ so the triangular equality is achieved again (the precise meaning of this assumption will be explained in proposition 5.5). Thus exponential and hyperbolic time consistent decision makers seem to exhibit the same condition.

After reinterpreting the triangular inequality uniquely on the basis of time perception (Observation 5.3), we will show that for subjective discounted utilities, time consistency is fully subsumed by the triangular equality property (Proposition 5.4).

Observation 5.3

For all ݐ ∈ ࣪ , ݅ ∈ ࣪, the Triangular Inequality can be rewritten as the comparison between the time distance and the difference between time perceptions for periods ݐ and ݅:

݀ሺ݅; ݐሻ ቆ < = > ቇ ߩሺݐሻ -ߩሺ݅ሻ ሺ13ሻ
Proposition 5.4

In the SDU model, decision makers are time consistent if and only if the triangular equality is verified ܥܶ ⟺ ݀ሺ݅; ݐሻ = ߩሺݐሻ -ߩሺ݅ሻ Proposition 5.4 confers to time consistent agents a common behavioral property in terms of time perception.

Time consistency clearly appears as a very restrictive condition since it imposes that the perceived gap between two periods is independent from the observational point. For example, if a year in one decade seems now very short then, after the mere passage of a decade, the coming year has to appear as short as the previous estimation (and obviously, DM exhibits time compression). Immediately, the combination of A4 and A6 induces the next observation.

Observation 5.5

The DM is time consistent if and only if, for all , ݐ ∈ ࣪ ሺ݇ ≤ ݐሻ, the initial time premium ߠ equals transmission time premium ߬ ܥܶ ⟺ ߠሺ,ݐ ݇ሻ = ߬ሺ݇, ݐሻ Indeed, Observation 5.5 will show that the time premium in A4 -that expresses, for instance, the additional delay needed to maintain indifference for postponed rewards -has to be equal to the time premium in A6 -that expresses, in this same case, the additional amount of time needed to preserve the indifference for postponed decision.

Notice that Observation 5.5 may be found independently from the SDU model. However, considering the subjective discounting axiom (A7) as a particular time discounting axiom for which ߠሺ,ݐ ݇ሻ = ߩ ିଵ ൫ߩሺݐሻ + ߩሺ݇ሻ൯ -ሺݐ + ݇ሻ, a key is introduced to understand the relationship between the standard interpretation of time consistency (based on time premiums which implicitly define discount rates) and the interpretation of time consistency (based on time perception) given by the subjective discounted utility model. This result may be alternatively presented with a more traditional means for intertemporal choices called Prelec's criterion. The Prelec's criterion ߛ ሺݐሻ is a useful measure of the nature of impatience for all discount functions8 . For convenience, we make the assumption for the following result that decision weights ݉ are twice differentiable in argument .ݐ Proposition 5.6

In the SDU model, he DM is time consistent if and only if the Prelec's criterion for any decision period ݅ is equal to the Prelec's criterion at the initial period

ܥܶ ⟺ ߛ ሺݐሻ = -ߩ ᇱᇱ ሺݐሻ ߩ ᇱ ሺݐሻ ⁄
Proposition 5.6 says that for hyperbolic agents, the decreasing impatience rate for time consistent hyperbolic Decision makers should be independent from the decision period. As a consequence, The Prelec's criterion might be used to dissociate two kinds of inconsistent decision makers: first, DM whose decreasing impatience rates (for given )ݐ are increasing function of decision period -namely ݅ < ݆ ⟹ ߛ ሺݐሻ < ߛ ሺݐሻ -and second, DM whose decreasing impatience rates (for given )ݐ are decreasing function of decision period -namely ݅ < ݆ ⟹ ߛ ሺݐሻ > ߛ ሺݐሻ. The first category of inconsistent decision makers reveals "classical" time inconsistency such as procrastination or temptation. The second category of decision makers induces an interesting but marginal studied case for which time inconsistency rises, for instance, when the DM exhibits perpetual sacrifice (the opposite case to procrastination) and never enjoys the pleasant task.

Section 6 Concluding Remarks

The intertemporal decision theory assumes a clear division of tasks between the exponential discounting, used to fit normative time consistent behavior, and hyperbolic discount functions, dedicated to observed time inconsistent behaviors. The Subjective Discounted Utility fully explains discounting by the nature of the DM's time perception, and as a consequence, time consistency can be interpreted with time distance properties. In this way, hyperbolic decision makers may be time consistent. Thus, time consistency has not exclusive concern with special discount function, but rather, is a matter of time perception transmission. This paper concludes by a claim in favor of a larger consideration for time perception in intertemporal choices and for the critical issue of time consistency, especially for long term perspectives as in retirement or environmental issues for which the time perception of a DM is likely to change. This claim is reinforced by considering the various fields of application connected to the literature on time consistency/inconsistency issue.

Appendices

Theorem 2.1 Observe that continuity holds for both payments and periods. First Order Separability (A2a) was first suggested by [START_REF] Koopmans | Representation of preference orderings over time[END_REF] and by [START_REF] Burness | On the Role of Separability Assumptions in Determining Impatience Implications[END_REF]. The utility representation (1) is directly obtained from [START_REF] Harvey | Proportional Discounting of future Costs and Benefits[END_REF] Function ݒ is continuous and strictly increasing with ܿ for all fixed ߬ ∈ ࣪, it is continuous with ߬ for all fixed ܿ ∈ ࣝ and ݒ ሺܿ * ; ߬ ሻ = 0. The intertemporal utility function ܷ is unique up to a positive transformation.

However, we cannot directly implement Harvey theorem (page 325) for our representative function since the latter has been built for only one decision period. That is why Second Order Separability was slightly amended to permit decision period independence. To understand the effect of A2b on the representation of time preferences, let us introduce the following lemma on multipliable separability for multi variable functions.

Lemma 1. Separability for a multi variable function

Given ݂ a function defined on the set ܶ , ݊ ∈ ℕ, ݃ defined on ܶ ିଵ , and

ℎ on ܶ, ∀ܽ ଵ , ܽ ଶ , … ܽ , ܾ ଵ , ܾ ଶ , … ܾ ∈ ܶ, ݂ሺܽ ଵ , ܽ ଶ , … , ܽ ିଵ , ܽ ሻ ݂ሺܽ ଵ , ܽ ଶ , … , ܽ ିଵ , ܾ ሻ = ݂ሺܾ ଵ , ܾ ଶ , … , ܾ ିଵ , ܽ ሻ ݂ሺܾ ଵ , ܾ ଶ , … , ܾ ିଵ , ܾ ሻ ⇔ ݂ሺܽ ଵ , ܽ ଶ , … ܽ ሻ = ݃ሺܽ ଵ , ܽ ଶ , … ܽ ିଵ ሻ × ℎሺܽ ሻ ሺ⇐ሻ is immediate ሺ⇒ሻ If ݂ሺܽ ଵ , ܽ ଶ , … , ܽ ିଵ , ܽ ሻ ݂ሺܽ ଵ , ܽ ଶ , … , ܽ ିଵ , ܾ ሻ ⁄ = ݂ሺܾ ଵ , ܾ ଶ , … , ܾ ିଵ , ܽ ሻ ݂ሺܾ ଵ , ܾ ଶ , … , ܾ ିଵ , ܾ ሻ ⁄ for all ܽ ଵ , ܽ ଶ , … ܽ , ܾ ଵ , ܾ ଶ , … , ܾ ∈ ܶ and setting ܾ ଵ = ܾ ଶ = ⋯ = ܾ ିଵ = ܾ , we have ݂ሺܽ ଵ , ܽ ଶ , … , ܽ ିଵ , ܽ ሻ = ݂ሺܽ ଵ , ܽ ଶ , … , ܽ ିଵ , ܾ ሻ ݂ሺܾ , ܾ , … , ܾ ሻ ⁄ × ݂ሺܾ , ܾ , … , ܾ , ܽ ሻ. Fix then ݃ሺܽ ଵ , ܽ ଶ , … , ܽ ିଵ ሻ = ݂ሺܽ ଵ , ܽ ଶ , … , ܽ ିଵ , ܾ ሻ ݂ሺܾ , ܾ , … , ܾ ሻ ⁄
and ℎሺܽ ሻ = ݂ሺܾ , ܾ , … , ܾ , ܽ ሻ, the desired expression follows.

We finally turn to the proof of our representation theorem.

  ሺ,ݔ ݏሻ ~ ሺ,ݕ ݐሻ ⟹ ሺ,ݔ ݏ + ݇ሻ ~ ሺ,ݕ ݐ + ݇ + ߠሻ For ݏ = 0, ሺ,ݔ 0ሻ ~ ሺ,ݕ ݐሻ ⟹ ሺ,ݔ ݇ሻ ~ ሺ,ݕ ݐ + ݇ + ߠሻ where ሺ,ݔ 0ሻ is the present equivalent to ሺ,ݕ ݐሻ. The standard model is the Exponential Discounted Utility (EDU) with ߠ = 0 and ߤ ሺݐሻ = ߣ ௧ and expresses constant impatience. EDU has been challenged in descriptive and prescriptive studies by Hyperbolic Discounted Utility (HDU) with ߠ = ݇ݐ and ߤ ሺݐሻ = 1 ሺ1 + ݐሻ ⁄ . Hyperbolic agents exhibit steeper discounting for time intervals closer to the present. To explain the previous paradox, we need to introduce the normative condition of time consistency.

Observation 4. 1

 1 Decision Weights of Theorem 2.1 are discount functions of time distance, that is, ∀݅ ∈ ࣪, ݐ∀ ∈ ࣪ , ݉ሺ݅; ݐሻ = ߤ൫݀ሺ݅; ݐሻ൯ ሺ6ሻ Observation 4.1 has many implications. It first says that dynamic intertemporal decision weights condense two distinct temporal postures: discounting and time perception. However, by doing this, no clear relation is established between discounting and time distance. Furthermore Observation 4.1 implies that time distance integrally subsumes the dynamical aspect of the model. As a consequence, time distance may be summarized by the combination of two complementary conditions: an initial condition which defines time perception for initial decision period and a transmission condition which determines how time perception is sensible to decision period. Initial condition: ݀ሺ0; ݐሻ = ߩ ሺݐሻ ≡ ߩሺݐሻ ሺ7ሻ Transmission condition: ݀ሺ݅; ݐሻ = ߩሺݐ -݅ሻ + ߮ሺ݅; ݐሻ ሺ8ሻ

  Estimation)∀݅, ݇ ∈ ࣪, ,ݏ∀ ݐ ∈ ࣪ , ∃߬ = ߬ሺ݅, ݇, ݐሻ: ࣪ × ࣪ × ࣪ → ࣪ such that: ሺ,ݔ ݏሻ ~ ሺ,ݕ ݐሻ ⟹ ሺ,ݔ ݏ + ݇ሻ ~ା ሺ,ݕ ݐ + ݇ + ߬ሻ A6 is a quite intuitive axiom. For ݏ = ݅, ሺ,ݔ ݅ሻ ~ ሺ,ݕ ݐሻ ⟹ ሺ,ݔ ݅ + ݇ሻ ~ା ሺ,ݕ ݐ + ݇ + ߬ሻ, the premium ߬ indicates how the coming time interval perception between periods ݅ and ݐ may be affected by the mere passage of time. Obviously if ߬ = 0 then time preservation holds. In return, if ߬ < 0 (resp. ߬ > 0) then the DM exhibits time compression (resp. extension) and time seems running faster (resp. more slowly) with age increasing. The next proposition will show that A6 is actually imposing a transmission condition for time distance, especially by clarifying the intimate relationship between ߬ and ߮.For a given time perception, the previous axiom fully determines the transmission condition associated to the time distance. Moreover, for all ݅ ∈ ࣪ and ݐ ∈ ࣪ , ߮ሺ݅; ݅ + ݐሻ = ݀ሺ݅; ݅ + ݐሻ -݀൫݅; ݅ + ݐ + ߬ሺ0, ݅, ݐሻ൯. Proposition 4.3 clearly shows that there exists a negative relation between ߬ and ߮ since when ߬ increases, (p2) from definition 5 implies that ݀ also increases and ߮ decreases. Notice that the magnitude of the "elasticity of ߮" to a change in ߬ depends on the nature of time perception. Under the light of proposition 4.3, we are now able to complete the message associated to proposition 4.2 by introducing the following corollary. AD) is the unique time distance satisfying objective time and time preservation ሺ * * ሻ Quasi Algebraic Distance (QAD) is the unique time distance satisfying objective time and time compression with premium ߬ = ݇ × ሺݐ -݅ሻ ሺ1 + ݅ሻ ⁄ ሺ * * * ሻ Logarithmic Time Distance (LTD) is the unique time distance satisfying logarithmic time perception and time preservation

  stationarity: 0ሻ ~ ሺ,ݕ ݐሻ ⟹ ሺ,ݔ ݇ሻ ~ ሺ,ݕ ݐ + ݇ሻ. The SDU Model reduces to the EDU model with ߤሺݐሻ = ߣ ௧ . With the same example, if ߩሺݐሻ = lnሺ1 + ݐሻ and logarithmic time perception induces ߩ ିଵ ൫ߩሺ1ሻ + ߩሺ2ሻ൯ = 5. The DM exhibits decreasing impatience and A7 becomes the central axiom used by[START_REF] Harvey | Proportional Discounting of future Costs and Benefits[END_REF], ሺ,ݔ 0ሻ ~ ሺ,ݕ ݐሻ ⟹ ሺ,ݔ ݇ሻ ~ ൫,ݕ ݐ + ݇ሺ1 + ݐሻ൯ with ߤሺݐሻ = ሺ1 + ݐሻ ି . By Observation 4.1, a direct consequence will be that the expression of dynamic decision weights changes for the general form ݉ሺ݅; ݐሻ = ߣ ௗሺ;௧ሻ . By analogy to the Subjective Expected Utility (SEU) for risky choices, the Subjective Discounted Utility is a two-stage-decision model in which the DM discounts subjective periods. In this way, the parameter ߣ can be meaningfully interpreted as a market discounting rule and potential individual deviations from this rule are fully explained by time distortions (time perception and/or time transmission) allowed by time distance functions. Theorem 5.1 (Subjective Discounted Utility) Assume (3) holds. Then the binary relation ≿ satisfies A7 if and only if, for all ሺܿ; ߬ሻ ∈ ∆ * , 0 < ߣ < 1, the intertemporal utility function is ܷܦܵ ሺܿ; ߬ሻ = ߣ ௗሺ;ఛ ೖ ሻ ݑሺܿ ሻ ା∞ ୀ ሺ11ሻ It is worth noticing that under (11), exponential decision makers are time consistent if and only if the algebraic distance (݀ሺ݅; ݐሻ = ݐ -݅) holds. Nevertheless, by theorem 4.6, hyperbolic agents may also be time consistent if and only if time distance is ݀ሺ݅; ݐሻ = ݎ ିଵ lnሺሺ1 + ݐሻ ሺ1 + ݅ሻ ⁄ ሻ. Then the Strotz's Theorem is a special case of time consistency in the Subjective Discounted Utility model.

Figure 1 .Figure 1

 11 Figure 1. The relation between the initial condition and the transmission condition in the SDU model

  decision weights may be rewritten as follows:∀ݐ ∈ ࣪ , ∀݅ ∈ ࣪ and for all ݇ ≥ 0, such that ݐ = ݅ + ݇, ݉ሺ݅; ݐሻ = ݉ሺ݅; ݅ + ݇ሻ = ݉ ሺ݇ሻ. The weight ݉ is defined on ࣪ since ݐ ≥ ݅ ⇒ ݇ ≥ 0 and by definition of ݉ , continuity of ݉ implies continuity of ݉ . Then it is easy to show that ݉ simultaneously verifies the three properties of definition 1. Note that ݉ሺ݅; ݅ሻ = 1 implies ݉ ሺ0ሻ = 1 ሺ݀1ሻ, ݉ሺ݅; ݐሻ > 0 gives ݉ ሺ݇ሻ > 0 ሺ݀2ሻ and impatience for ݉ induces decreasing ݉ with ݇ ሺ݀3ሻ. Lastly ሺ݀1ሻ, ሺ݀2ሻ and ሺ݀3ሻ constrain the image set to ሿ0; 1ሿ and then ݉ሺ݅; ݐሻ = ߤ ሺݐ -݅ሻ ∎ Proposition 3.2 By (1), ݉ሺ݅; ݐሻ = ߤ ሺݐሻ ߤ ሺ݅ሻ ⁄ directly implies time consistency. So assume time consistency holds and consider two dated rewards ሺ,ݔ ݏሻ and ሺ,ݕ ݐሻ ൫,ݏ ݐ ∈ ࣪ , ݅, ݆ ∈ ࣪൯ such that ሺ;ݔ ݏሻ~ሺ;ݕ ݐሻ. Then by (1) we have ݉ሺ݅; ݐሻ ݉ሺ݅; ݏሻ ⁄ = ݑሺݔሻ ݑሺݕሻ ⁄ and by time consistencyሺ;ݔ ݏሻ~ሺ;ݕ ݐሻ that is ݉ሺ݆; ݐሻ ݉ሺ݆; ݏሻ ⁄ = ݑሺݔሻ ݑሺݕሻ ⁄ . Setting ݆ = ݏ and ݅ = 0 gives ݉ሺ݆; ݐሻ = ߤ ሺݐሻ ߤ ሺ݆ሻ ⁄ ∎ Corollary 3.3 That multiplicative separability induces time consistency is straightforward. Reciprocally, assume time consistency then by proposition 3.2 ݉ሺ݅; ݐሻ ݉ሺ݅; ݏሻ ⁄ = ݉ሺ݆; ݐሻ ݉ሺ݆; ݏሻ ⁄ ൫,ݏ ݐ ∈ ࣪ , ݅, ݆ ∈ ࣪൯. By Lemma 1 wehave ݉ሺ݅; ݐሻ = ߮ሺ݅ሻ × ߴሺݐሻ. Moreover consistent weight can be written as ݉ሺ݅; ݐሻ = ߤ ሺݐሻ ߤ ሺ݅ሻ ⁄ and by association ߴሺݐሻ = 1 ߮ሺݐሻ ⁄ = ߤ ሺݐሻ. As a consequence, ݉ሺ݅; ݐሻ = ݉ሺ݅; ݏሻ × ݉ሺ;ݏ ݐሻ. Turn to second part of Corollary 3.3. For all ሺ;ݔ ߬ሻ ∈ ∆ ሺ݅, ݆ ∈ ࣪, ݅ < ݆ሻ by (1) we have ܷ ሺ;ݔ ݐሻ = ∑ ݉ሺ݅; assume time consistency holds. Stop Watch Time implies ݉ሺ݅; ݅ + ݐሻ = ݉ሺ0; ݐሻ or ݉ሺ݅; ݐሻ = ݉ሺ0; ݐ -݅ሻ. Time consistency implies ݉ሺ݅; ݐሻ = ݉ሺ0; ݐሻ ݉ሺ0; ݅ሻ ⁄ and by SWT ߤ ሺݐ -݅ሻ = ߤ ሺݐሻ ߤ ሺ݅ሻ ⁄ , functional equation whose broadest solution is ߤ ሺݐሻ = ߣ ௧ . Conversely, if we retain EDU then ݉ሺ0; ݐሻ = ߣ ௧ and SWT implies ݉ሺ݅; ݅ + ݐሻ = ߣ ௧ then ݉ሺ݅; ݐሻ = ߣ ௧ି = ݉ሺ0; ݐሻ ݉ሺ0; ݅ሻ ⁄ . ∎ Observation 4.1 First note that ݉ሺ݅; ݐሻ = ߤ ሺݐ -݅ሻ = ߤ൫݂ ሺݐ -݅ሻ൯. We have to show that ݂ ሺݐ -݅ሻ = ߩ ሺݐ -݅ሻ by transposing properties of ݉ to ݂ and by verifying that the three properties of definition 2 hold. It is quite easy and the end of the proof is left to reader. ∎ Proposition 4.2 The proof focuses on uniqueness. Assume ݀ሺ݅; ݐሻ = ߩ ሺݐ -݅ሻ is given then setting ݅ = 0 associates to ݀ሺ0; ݐሻ a unique correspondent time perception ߩሺݐሻ. Therefore we have also a unique time premium ߮ such that ߮ሺ݅, ݇, ݐሻ = ݀ሺ݅; ݐሻ -ߩሺݐ -݅ሻ. ∎ Proposition 4.3 First A6 is equivalent to ݀ሺ݅; ݐሻ = ݀ሺ݅ + ݇; ݐ + ݇ + ߬ሻ so for ݅ = 0 we have ߩሺݐሻ = ݀ሺ݇; ݐ + ݇ + ߬ሻ and for ݇ = ݅ the condition becomes ߩሺݐሻ = ݀ሺ݅; ݐ + ݅ + ߬ሻ = ߩ ሺݐ + ߬ሻ. Obviously, ߩ ሺݐ + ߬ሻ = ߩሺݐሻ is a transmission condition. We now determine the connection between ߮ and ߬. By A6 ൫ሺ,ݔ 0ሻ~ሺ,ݕ ݐሻ ⇒ ሺ,ݔ ݅ሻ~ሺ,ݕ ݅ + ݐ + ߬ሻ൯ and by (1) we deduce ߩሺݐሻ = ݀൫݅; ݅ + ݐ + ߬ሺ݅; ݐሻ൯. By transmission condition, ߩሺݐሻ = ݀ሺ݅; ݅ + ݐሻ -߮ሺ݅; ݅ + ݐሻ then ߮ሺ݅; ݅ + ݐሻ = ݀ሺ݅; ݅ + ݐሻ -݀൫݅; ݅ + ݐ + ߬ሺ݅; ݐሻ൯ and for ݅ = 0, ߮ሺ0; ݐሻ = ߩሺݐሻ -ߩ൫ݐ + ߬ሺ0; ݐሻ൯. ∎ Corollary 4.4 By proposition 4.2, a time distance is uniquely determined by a combination of an initial condition and a transmission condition. Then we only need to focus on the necessary condition. For algebraic distance the proof is immediate. Turn to quasi algebraic distance and start from the equality ݀ሺ݅; ݏሻ = ݀ሺ݅ + ݇; ݏ + ݇ + ߬ሻ given by A6. Setting ݅ = 0 and by objective time, ݀ሺ݇; ݏ + ݇ + ߬ሻ = ݏ and by the assumption ߬ = ݇ × ሺݐ -݅ሻ ሺ1 + ݅ሻ ⁄ , we have ݀ሺ݇; ݏ + ݇ + ݇ݏሻ = ݀ሺ݇; ݇ + ሺ1 + ݇ሻݏሻ = .ݏ With the notation ݀ሺ݇; ݇ + ሺ1 + ݇ሻݏሻ = ݃ ൫ሺ1 + ݇ሻݏ൯ and setting ݐ = ሺ1 + ݇ሻ,ݏ the following equality is obtained ݃ ሺݐሻ = ݐ ሺ1 + ݇ሻ ⁄ . Last for ݇ = ݅, ݀ሺ݅; ݅ + ݐሻ = ݐ ሺ1 + ݅ሻ ⁄ or ݀ሺ݅; ݐሻ = ሺݐ -݅ሻ ሺ1 + ݅ሻ ⁄ . The third part of corollary 4.3 concerning logarithmic time distance function is immediate. ∎ Observation 4.5 Assume time preservation holds then for ݐ ∈ ࣪ ሺ݅ ∈ ࣪ሻ, ݀ሺ݅; ݐሻ = ݀ሺ0; ݐ -݅ሻ. Hence by (1) and observation 4.1 ݉ሺ݅; ݐሻ = ݉ሺ0; ݐ -݅ሻ must hold and SWT is verified. The demonstration for the reciprocal implication will be proved by the same way. ∎ Theorem 4.6 By Observation 4.1 ݉ሺ݅; ݐሻ = ߤ ∘ ݀ሺ݅; ݐሻ and by Proposition 3.2 time consistency holds if and only if ݉ሺ݅; ݐሻ = ߤ ሺݐሻ ߤ ሺ݅ሻ ⁄ . As a consequence the DM is time consistent iff ߤ ∘ ݀ሺ݅; ݐሻ = ߤ ሺݐሻ ߤ ሺ݅ሻ ⁄ where ߤ and ߤ are two discount functions not necessarily equal. Observe that as ߤ is continuous and strictly decreasing over ࣪, then ߤ ିଵ does exist. The remaining demonstration will prove that the function ߨሺ݅; ݐሻ = ߤ ିଵ ሺߤ ሺݐሻ ߤ ሺ݅ሻ ⁄ ሻ is a time distance function. By ሺ݀1ሻ in definition 1, ߤ ିଵ ∶ ሿ0; 1ሿ → ሾ0; +∞ሾ. As ߤ satisfies ሺ݀2ሻ then ߤ ሺݐሻ ߤ ሺ݅ሻ ⁄ > 0and by impatience ݐ ≥ ݅ implies ߤ ሺݐሻ ߤ ሺ݅ሻ ⁄ ≤ 1. As a result ߨ ∶ ࣪ × ࣪ → ሾ0; +∞ሾ. Furthermore ߤ is continuous over ࣪ = ሾ0; +∞ሾ so over ࣪ ⊂ ࣪, ߤ ିଵ is continuous over ሿ0; 1ሿ therefore ߨ is continuous over ࣪ × ࣪ . By definition of ߤ ିଵ , we have ߨሺ݅; ݐሻ ≥ 0 ሺ1ሻ and ߨ is increasing with ݐ since ↑ ݐ ⇒ ߤ ሺݐሻ ↓ . Lastly ߨሺ݅; ݅ሻ = ߤ ିଵ ሺߤ ሺ݅ሻ ߤ ሺ݅ሻ ⁄ ሻ = ߤ ିଵ ሺ1ሻ = 0 ሺ3ሻ. Observe that time distance ݀ሺ݅; ݐሻ = ߤ ିଵ ሺߤ ሺݐሻ ߤ ሺ݅ሻ ⁄ ሻ is not unique since no constraint is imposed on ߤ. obtain ߤ൫ߩሺݐሻ൯ = ߣ ௧ we restate a theorem by[START_REF] Lapied | A Subjective Discounted Utility Model[END_REF]. According to Theorem 2.1, axioms A.1 to A.3 induce the functional representation (3). Therefore if ሺ,ݔ ݏሻ ~ ሺ,ݕ ݐሻ then A7 implies for all ,ݏ ,ݐ ݇ ∈ ࣪, ߤ൫ߩሺݐሻ൯ × ߤ൫ߩሺݏሻ + ߩሺ݇ሻ൯ = ߤ൫ߩሺݏሻ൯ × ߤ൫ߩሺݐሻ + ߩሺ݇ሻ൯. Hence choosing ߩሺ݇ሻ = ߩሺݐሻ -ߩሺݏሻ, we obtain ߩሺ݇ሻ൯ቁ. Since ݂ሺݐሻ = ln ቀߤ൫ߩሺݐሻ൯ቁ is continuous then the previous functional equation is a Jensen's equation whose broadest solution is ln ቀߤ൫ߩሺݐሻ൯ቁ = ܽߩሺݐሻ + ܾ.

  ሺ,ݔ ݏሻ ≿ ሺ,ݕ ݐሻ means

	ሺܿ * intertemporal utility function ܷ on ∆ and then FOS allows the restriction of the analysis to instantaneous utility ௦ ;ݔ ߬ሻ ≿ ሺܿ * ௧ ;ݕ ߬ሻ. The proof of next theorem 2.1 will show that A1 implies the existence of an functions ݒ ሺ,ݔ ݐሻ defined on ࣝ × ࣪ .
	A2b (Second Order Separability)
	For all ,ݔ ,ݕ ݖ ∈ ࣝ, ߬ ∈ ॻ and ݅, ݆ ∈ ࣪ ሺ݅ < ݆ሻ, If ሺ,ݔ ݏሻ ~ ሺ,ݕ ݐሻ ሺ,ݔ ′ݏሻ ~ ሺ,ݕ ′ݐሻ then ሺܿ * ௦ ;ݖ ߬ሻ ∼ ൫ܿ * ௦,௧ ,ݔ ;ݕ ߬൯ ⟹ ሺܿ * ௦ᇱ ;ݖ ߬ሻ ∼ ൫ܿ * ௦ᇱ,௧ᇱ ,ݔ ;ݕ ߬൯

  A11 and A12 are very standard axioms and insure the existence of a representative intertemporal utility function.

	A1. ∀݅ ∈ ࣪, ≿ is a regular binary relation if and only if ≿ satisfies the three following axioms and the existence of a neutral element. A11. ∀݅ ∈ ࣪, ≿ is a weak order (complete and transitive binary relation) A12. ∀݅ ∈ ࣪, ≿ is monotonic ∀݇ ∈ ܵ, ܿ ≥ ܿ′ and ∃݇ ∈ ܵ, ܿ > ܿ′ then ሺܿ; ߬ሻ ≻ ሺܿ′; ߬ሻ A13. ∀݅ ∈ ࣪, ≿ is continuous ∀݇ ∈ ܵ, ∃ܰ > 0 such that for all ߜ , 0 < ߜ < ܰ , we have: ሺܿ; ߬ሻ ≻ ሺܿ′; ߬′ሻ ⟹ ሺܿ; ߬ሻ ≻ ሺܿ′ , … , ܿ′ , … ; ߬′ , … , ߬′ + ߜ , … ሻ ሺܿ; ߬ሻ ≻ ሺܿ′; ߬′ሻ ⟹ ሺܿ; ߬ሻ ≻ ሺܿ′ , … , ܿ′ + ߜ , … ; ߬′ , … , ߬′ , … ሻ

  ∀݅, ݆ ∈ ࣪ ሺ݅ < ݆ሻ, choose ݔ and ݕ such that ሺܿ * ௦ ;ݔ ߬ሻ ~ ሺܿ * ௧ ;ݕ ߬ሻ and ሺܿ * ௦ᇱ ;ݔ ߬′ሻ ~ ሺܿ * ௧ᇱ ;ݕ ߬′ሻ. Then by (1), ݒ ሺ,ݔ ݏሻ = ݒ ሺ,ݕ ݐሻ and ݒ ሺ,ݔ ′ݏሻ = ݒ ሺ,ݕ ′ݐሻ.

	Moreover take ݖ ∈ ࣝ such that ሺܿ * ݒ ሺ,ݖ ݏሻ ݒ ሺ,ݔ ݏሻ ⁄ = ݒ ሺ,ݖ ′ݏሻ ݒ ሺ,ݔ ′ݏሻ	௦ ;ݖ ߬ሻ ~ ൫ܿ *	௦,௧ ,ݔ ;ݕ ߬൯ and by A2b ሺܿ *	௦ᇱ ;ݖ ߬′ሻ ~ ൫ܿ *	௦ᇱ,௧ᇱ ,ݔ ;ݕ ߬′൯. Hence

⁄ and by lemma 1 ݒ ሺ,ݔ ݏሻ = ݉ሺ݅; ݏሻ × ݑሺݔሻ with ݉ሺ݅; ݏሻ = ݒ ሺ݅, ݏሻ ݒ ሺ݅, ݅ሻ ⁄

and ݑሺݔሻ = ݒ ሺ,ݔ ݅ሻ. Then ݉ሺ݅; ݅ሻ = 1 and by monotonicity ݉ሺ݅; ݐሻ > 0. Hence by A3, (*) implies ݑሺ0ሻ = 0 and notice that

[START_REF] Harvey | Proportional Discounting of future Costs and Benefits[END_REF]

insures that ݑ is unique up to a positive linear transformation and ݑ increases with ݔ induces ݑሺݔሻ > 0 for ݔ > 0. Lastly, (**) implies that ݉ is decreasing with ݐ and (***) that ݉ is increasing with ݅. ∎

The precise definition of a regular preference will be provided in Appendices.

For models that integrate some complementarity between successive periods, see Choquet Utility Models as in[START_REF] Gilboa | Expectation and variation in Multi-period decisions[END_REF],de Waegenaere and Wakker (2001) or[START_REF] Chateauneuf | Some characterizations of non-additive multi-period models[END_REF].

Section 5 develops this point by introducing the Prelec's criterion which is a measure for decreasing impatience degree.

Actually time perception needs to be linear, that is ߩሺݐሻ = ܽ × ݐ with ܽ > 0 but it is easily to show that in the SDU model, linear time perceptions may be reduced to the case of objective time.

[START_REF] Prelec | Decreasing impatience: A criterion for non-stationary time preference and 'hyperbolic' discounting[END_REF] proposes that the convexity of the log of the discount function ݂ሺݐሻ = ln ߤሺݐሻ is a measure of the impatience degree with ߛ ሺݐሻ = -݂′′ሺݐሻ ݂′ሺݐሻ ⁄ .

As ߤሺ0ሻ = 1 and ߩሺ0ሻ = 0 so ܾ = 0. Hence ln൫ߤሺݐሻ൯ = ܽߩሺݐሻ with ܽ < 0 since ߤ is a discount function. Finally we obtain ߤሺ݆ሻ = ߣ ఘሺሻ ሺߣ = ݁ ି , ݎ = -ܽሻ. Conversely, the SDU verifies (3) so according to Theorem 2.1, A.1 to A.3 are implied. In addition, if for all ,ݐ ݇ ∈ ࣪ ߤሺݐሻ = ߣ ఘሺ௧ሻ hence ߤ൫ߩሺݐሻ + ߩሺ݇ሻ൯ = ߣ ఘሺ௧ሻାఘሺሻ then under (3) we have ሺ,ݔ ݏሻ ~ ሺ,ݕ ݐሻ ⟹ ቀ,ݔ ߩ ିଵ ൫ߩሺݏሻ + ߩሺ݇ሻ൯ቁ ~ ቀ,ݕ ߩ ିଵ ൫ߩሺݐሻ + ߩሺ݇ሻ൯ቁ. Since ߤሺݐሻ = ߣ ௧ , then by identification, ߤ൫݀ሺ݅; ݐሻ൯ = ߣ ௗሺ;௧ሻ immediately follows. ∎ Corollary 5.2 The proof of Corollary 5.2 is immediate. However, to clarify the core meaning of this corollary, assume that decision weights ݉ are twice differentiable in .ݐ Then Corollary 5.2 is illustrated by the use of the Prelec's criterion (2004). Recall that the Prelec's criterion is measure of decreasing impatience for all discount functions. We show that the Prelec's criterion in the SDU model only depends on time distance derivatives. The Dynamic Prelec's criterion for decision period

. By theorem 5.1, the Prelec's criterion in the SDU model is reduced to the simpler following expression

. Then the nature of impatience is integrally determined by the time distance properties. ∎ Observation 5.3 For ,ݏ ݐ ∈ ࣪ , ݏ < ݐ and ݅ ∈ ࣪, the triangular equality ݀ሺ݅; ݐሻ = ݀ሺ݅; ݏሻ + ݀ሺ;ݏ ݐሻ is a Sincov functional equation and by [START_REF] Aczél | Lectures on Functional Equations and Their Applications[END_REF], the broadest solution of such an equation is ݀ሺ݅; ݐሻ = ℎሺݐሻ -ℎሺ݅ሻ = ℎሺݐሻ -ℎሺ0ሻ -ℎሺݐሻ + ℎሺ0ሻ = ߩሺݐሻ -ߩሺ݅ሻ. For the triangular inequality, ݀ሺ݅; ݐሻ > ݀ሺ݅; ݏሻ + ݀ሺ;ݏ ݐሻ implies ݀ሺ;ݏ ݐሻ < ݀ሺ݅; ݐሻ -݀ሺ݅; ݏሻ and for ݅ = 0 we have ݀ሺ;ݏ ݐሻ < ߩሺݐሻ -ߩሺݏሻ. The reversed triangular inequality implies the opposite inequality. ∎ Proposition 5.4 By proposition 3.2, time consistency implies ݉ሺ݅; ݐሻ = ߤ ሺݐሻ ߤ ሺ݅ሻ ⁄ which can be rewritten in the SDU approach as ߣ ௗሺ;௧ሻ = ߣ ఘሺ௧ሻିఘሺሻ and the triangular equality holds. The reciprocal implication is straightforward. ∎


Pierre Gabriel
email: gabriel@ann.jussieu.fr

The Shape of the Polymerization Rate in the Prion Equation

Keywords: Aggregation-fragmentation equations, self-similarity, eigenproblem, size repartition, polymerization process, data fitting, inverse problem AMS Class. No. 35F50, 35Q92, 35R30, 45C05, 45K05, 45M99, 45Q05, 65D10, 92D25

, where the fragmentation dependence on prion strains is investigated. Our improvement is to use power laws for the rates, whereas [30] used a constant polymerization coefficient and linear fragmentation.

Introduction

Polymerization of prion proteins is a central event in the mechanism of prion diseases. The cells naturally produce the normal form of this protein (PrPc), but there exists a pathogenic form (PrPsc) present in infected cells as polymers. These polymers have the ability to transconform and attach the normal form by a polymerization process not yet very well understood. Mathematical modelling is a promising tool to investigate this mechanism, provided we can find accurate parameters. We consider the following model of prion proliferation (see [START_REF] Calvez | Prion dynamic with size dependency -strain phenomena[END_REF][START_REF] Calvez | Size distribution dependence of prion aggregates infectivity[END_REF][START_REF] Greer | A mathematical analysis of the dynamics of prion proliferation[END_REF][START_REF] Laurençot | Well-posedness for a model of prion proliferation dynamics[END_REF]):

                   dV (t) dt = λ -V (t) δ + ∞ 0 τ (x)u(x, t) dx , ∂ ∂t u(x, t) = -V (t) ∂ ∂x τ (x)u(x, t) -[µ 0 + β(x)]u(x, t) + 2 ∞ x
β(y)κ(x, y) u(y, t) dy, u(0, t) = 0, u(x, 0) = u 0 (x).

(

) 1
This equation is the continuous version of the discrete model of [START_REF] Masel | Quantifying the kinetic parameters of prion replication[END_REF] (see [START_REF] Doumic | Scaling limit of a discrete prion dynamics model[END_REF] for the derivation).

System [START_REF] Calvez | Prion dynamic with size dependency -strain phenomena[END_REF] is a quite general model able to reproduce biological experimental results (see [START_REF] Lenuzza | Modélisation de la réplication des Prions: implication de la dépendance en taille des agrégats de PrP et de l'hétérogénéité des populations cellulaires[END_REF]). In this equation, V (t) is the quantity of PrPc which is produced by the cell with a rate λ, and degraded with the rate δ. The quantity of aggregates of PrPsc with size x at time t is denoted by u(x, t). Polymers of PrPsc can attach monomers (PrPc) with rate τ and can split themselves in two smaller polymers with rate β. To fit with experimental observations [START_REF] Silveira | The most infectious prion protein particles[END_REF], the polymerization and fragmentation rates should depend on the polymer size x (see [START_REF] Calvez | Prion dynamic with size dependency -strain phenomena[END_REF][START_REF] Calvez | Size distribution dependence of prion aggregates infectivity[END_REF]). When fragmentation occurs, a polymer of size y produces two pieces of size x and y -x with the probability κ(x, y). The natural assumptions (see [START_REF] Michel | General entropy equations for structured population models and scattering[END_REF][START_REF] Michel | General relative entropy inequality: an illustration on growth models[END_REF][START_REF] Perthame | Transport equations in biology[END_REF][START_REF] Laurençot | Well-posedness for a model of prion proliferation dynamics[END_REF]) on the kernel κ are then

Supp κ ⊆ {(x, y), 0 ≤ x ≤ y} (2)
since fragmentation produces smaller aggregates than the initial one,

∀y > 0, κ(x, y) dx = 1 (κ(•, y) is a probability measure) (3)
and ∀x ≤ y, κ(x, y) = κ(y -x, y) (4)
because of the indistinguishability of the two pieces of sizes x and y -x after fragmentation. A consequence of the assumptions ( 2)-( 4) is that the mean size of the fragments obtained from a polymer of size y is y/2 :

∀y > 0, xκ(x, y) dx = y 2 . (5)
In the discrete model of [START_REF] Masel | Efficient inhibition of prion replication by prp-fc2 suggests that the prion is a prpsc oligomer[END_REF][START_REF] Masel | Quantifying the kinetic parameters of prion replication[END_REF][START_REF] Masel | Designing drugs to stop the formation of prion aggregates and other amyloids[END_REF], a critical size n of polymers is considered. Under this size, the polymers are unstable, so the proteins that compose them go back to their normal form PrPc. This effect is taken into account in the continuous model of [START_REF] Greer | A mathematical analysis of the dynamics of prion proliferation[END_REF][START_REF] Greer | Effects of general incidence and polymer joining on nucleated polymerization in a model of prion proliferation[END_REF] with the critical size x 0 . As a consequence, there is one more integral term of apparition of monomers in the equation on V. Nevertheless, even if n > 0 in the discrete model, the value of x 0 can be taken equal to 0 in the continuous one, as justified in [START_REF] Doumic | Scaling limit of a discrete prion dynamics model[END_REF]. Then the fact that small polymers are unstable, and so do not exist, is taken into account by the boundary condition u(0, t) = 0. The parameter µ 0 is a death term which will be discussed later.

In the article [START_REF] Zampieri | Investigating the conformational stability of prion strains through a kinetic replication model[END_REF], model [START_REF] Calvez | Prion dynamic with size dependency -strain phenomena[END_REF] with the specific parameters of [START_REF] Engler | Analysis of a model for the dynamics of prions ii[END_REF][START_REF] Greer | A mathematical analysis of the dynamics of prion proliferation[END_REF][START_REF] Greer | Effects of general incidence and polymer joining on nucleated polymerization in a model of prion proliferation[END_REF][START_REF] Masel | Efficient inhibition of prion replication by prp-fc2 suggests that the prion is a prpsc oligomer[END_REF][START_REF] Masel | Quantifying the kinetic parameters of prion replication[END_REF][START_REF] Masel | Designing drugs to stop the formation of prion aggregates and other amyloids[END_REF] is used. It assumes that the polymerization rate does not depend on polymer size (τ (x) ≡ τ 0 ) and the fragmentation is described by β(x) = β 0 x and κ(x, y) = 1l 0≤x≤y 1 y . In this case, equation (1) can be reduced to a nonlinear system of ODEs (see [START_REF] Engler | Analysis of a model for the dynamics of prions ii[END_REF][START_REF] Greer | A mathematical analysis of the dynamics of prion proliferation[END_REF][START_REF] Greer | Effects of general incidence and polymer joining on nucleated polymerization in a model of prion proliferation[END_REF] or even [START_REF] Masel | Efficient inhibition of prion replication by prp-fc2 suggests that the prion is a prpsc oligomer[END_REF][START_REF] Masel | Quantifying the kinetic parameters of prion replication[END_REF][START_REF] Masel | Designing drugs to stop the formation of prion aggregates and other amyloids[END_REF]). This system is used in [START_REF] Zampieri | Investigating the conformational stability of prion strains through a kinetic replication model[END_REF] to investigate the parameters involved in the so-called "prion strain phenomenon". This phenomenon is the ability of the same encoded protein to encipher a multitude of phenotypic states. Prion strains are characterized by their incubation period (time elapsed between experimental inoculation of the PrPsc infectious agent and clinical onset of the disease) and their lesion profiles (patterns produced by the death of neurones) in the brain [START_REF] Fraser | Scrapie in mice : Agent-strain differences in the distribution and intensity of grey matter vacuolation[END_REF]. A reason for the strain diversity can be that there exist various conformation states of PrPsc, namely various misfoldings of the prion protein, which lead to various biological and biochemical properties [START_REF] Morales | The prion strain phenomenon: Molecular basis and unprecedented features[END_REF][START_REF] Rezaei | Amyloidogenic unfolding intermediates differentiate sheep prion protein variants[END_REF]. These different conformations could be characterizable by their different stability against denaturation. Indeed, recent studies on prion disease have confirmed that the incubation time is related not only to the quantity of pathogenic protein inoculated ( u 0 (x) dx) and the level of expression of PrPc (λ), but also to the resistance of prion strains against denaturation [START_REF] Legname | Continuum of prion protein structures enciphers a multitude of prion isolate-specified phenotypes[END_REF] in terms of the concentration of guanide hydrochloride (Gdn-HCl) required to denature 50% of the diseasecausing protein. Other studies have highlighted a strong relationship between the stability of the prion protein against denaturation and neuropathological lesion profiles [START_REF] Legname | Strain-specified characteristics of mouse synthetic prions[END_REF][START_REF] Sigurdson | Prion strain discrimination using luminescent conjugated polymers[END_REF]. A natural question is then to know how the conformation states of the PrPsc protein affect the parameters of model ( 1) and more precisely which of these parameters is the most strain dependent. One of the conclusions of [START_REF] Zampieri | Investigating the conformational stability of prion strains through a kinetic replication model[END_REF] is that the key parameter describing strain-dependent replication kinetics is the fibril breakage rate β 0 . Using the model, some coefficients already defined in [START_REF] Masel | Quantifying the kinetic parameters of prion replication[END_REF] are calculated from experimental data, and the dependence between them indicates that the fragmentation rate β 0 is the most adapted parameter to be strain dependent. But, in this framework, the fragmentation dependence is not sufficient to fit the data precisely, and the polymerization rate τ 0 is also assumed to change with the strain. In particular, to keep a one-parameter family of models, the polymerization rate is assumed to depend on the fragmentation one through a power relation:

∃φ s.t. τ 0 = β 0 φ . (6)
However, polymerization and fragmentation are two different biochemical processes which have no reason to be linked together.

In this paper we prove that we can get rid of such a relation by considering a larger class of parameters for model [START_REF] Calvez | Prion dynamic with size dependency -strain phenomena[END_REF]. We take advantage of a general framework developed in [START_REF] Calvez | Prion dynamic with size dependency -strain phenomena[END_REF][START_REF] Calvez | Size distribution dependence of prion aggregates infectivity[END_REF][START_REF] Laurençot | Well-posedness for a model of prion proliferation dynamics[END_REF], where coefficients can be more general than in [START_REF] Engler | Analysis of a model for the dynamics of prions ii[END_REF][START_REF] Greer | A mathematical analysis of the dynamics of prion proliferation[END_REF][START_REF] Greer | Effects of general incidence and polymer joining on nucleated polymerization in a model of prion proliferation[END_REF][START_REF] Masel | Efficient inhibition of prion replication by prp-fc2 suggests that the prion is a prpsc oligomer[END_REF][START_REF] Masel | Quantifying the kinetic parameters of prion replication[END_REF][START_REF] Masel | Designing drugs to stop the formation of prion aggregates and other amyloids[END_REF][START_REF] Zampieri | Investigating the conformational stability of prion strains through a kinetic replication model[END_REF], and we consider power laws as in [START_REF] Escobedo | Dust and self-similarity for the Smoluchowski coagulation equation[END_REF][START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF][START_REF] Laurençot | Well-posedness for a model of prion proliferation dynamics[END_REF][START_REF] Michel | Existence of a solution to the cell division eigenproblem[END_REF]. Thanks to this extension, we find a shape for the polymerization rate allowing us to fit experimental data with only the fragmentation rate dependent on strain. The experimental data we use are the stability against denaturation G and the growth rate of the quantity of PrPsc (r), which is linked to the incubation time. These data are strain dependent, and we can find in [START_REF] Zampieri | Investigating the conformational stability of prion strains through a kinetic replication model[END_REF] estimated empirical values for different strains. In Section 2, we present the class of coefficients we consider, and give the necessary assumptions for the mathematical investigations. Most of these assumptions are already made in [START_REF] Masel | Quantifying the kinetic parameters of prion replication[END_REF][START_REF] Zampieri | Investigating the conformational stability of prion strains through a kinetic replication model[END_REF] and, thanks to them, we are able to link r and G to the mathematical model. In Section 3 we give the dependence of these parameters on the polymerization and fragmentation rates. Then we give a relation between them using only the fragmentation dependency on strain and are able to fit data with an accurate shape of the polymerization rate.

Assumptions and Method

To avoid the use of a power relation like [START_REF] Engler | Analysis of a model for the dynamics of prions ii[END_REF] between the polymerization and the fragmentation, we allow the parameters τ and β to depend on the size x with power laws :

∃γ, ν and β 0 , τ 0 s.t., ∀x > 0, β(x) = β 0 x γ and τ (x) = τ 0 x ν . ( 7
)
Concerning the fragmentation kernel κ, we assume that there exists a measure κ 0 on [0, 1] such that

∀x, y > 0, κ(x, y) = 1 y κ 0 x y . (8)
Then the natural assumptions on κ become

1 0 κ 0 (z) dz = 1 and κ 0 (z) = κ 0 (1 -z). (9)
We assume that the exponents ν, γ, the kernel κ 0 and the death rate µ 0 are strain independent unlike the "intensities" β 0 and τ 0 . First, we look which of β 0 and τ 0 is the most characteristic of the strain.

Then we find a value of ν allowing to fit experimental data as well as [START_REF] Zampieri | Investigating the conformational stability of prion strains through a kinetic replication model[END_REF] but without assumption (6). We will see in Section 3 that the method naturally gives the parameter ν, because it turns out that γ and κ 0 do not appear in the final argument.

The method proposed in [START_REF] Masel | Quantifying the kinetic parameters of prion replication[END_REF][START_REF] Zampieri | Investigating the conformational stability of prion strains through a kinetic replication model[END_REF] is to investigate the dynamics of the system (1) linearized at the disease-free equilibrium. Then we can use the eigenelements as suggested in [START_REF] Calvez | Prion dynamic with size dependency -strain phenomena[END_REF][START_REF] Calvez | Size distribution dependence of prion aggregates infectivity[END_REF]. To make this approximation valid, we make some classical assumptions. First we assume that, in the biological experiments (namely before animal death), the amount of PrPc in cells remains close to the sane value V = λ δ (obtained by setting u ≡ 0 in the first equation of ( 1)). Quantity V being assumed to be constant, we consider the eigenelements r and U of the equation for polymers associated with V :

       rU (x) = - V ∂ ∂x τ (x)U (x) -[µ 0 + β(x)]U (x) + 2 ∞ x β(y)κ(x, y) U (y) dy, U (0) = 0, U (x) dx = 1. ( 10
)
The principal eigenvalue r represents the growth rate of the solutions, and the eigenvector U is the corresponding size distribution of polymers. To consider that the size distribution is close to this profile along the experiments, we assume that the initial value u 0 (x) is already close to U . It is a reasonable assumption if contaminations are made from other infected individuals. Then the solution of ( 1) is reduced to

V (t) ≃ V and u(x, t) ≃ u 0 (y)dy U (x)e rt . (11)
To solve our problem, we use the idea that the eigenelements r and U change with the prion strains (see [START_REF] Calvez | Size distribution dependence of prion aggregates infectivity[END_REF][START_REF] Lenuzza | Modélisation de la réplication des Prions: implication de la dépendance en taille des agrégats de PrP et de l'hétérogénéité des populations cellulaires[END_REF] for instance). Because present experiments do not allow one to measure the size distribution U , we link it to the stability against denaturation G by the relation

∃a > 0, b s.t. G = ax + b (12)
where x = x U (x) dx is the mean size of the polymers since U (x) dx = 1. Such an affine dependence, suggested in [START_REF] Zampieri | Investigating the conformational stability of prion strains through a kinetic replication model[END_REF], is based on the idea that the longer x is, the larger G will be. In [START_REF] Legname | Strain-specified characteristics of mouse synthetic prions[END_REF][START_REF] Zampieri | Investigating the conformational stability of prion strains through a kinetic replication model[END_REF] we can find estimated empirical values of G (as a concentration, in mol, of Gnd-HCL required to denature 50% of PrPsc) and of the growth rate r (computed from measures of incubation times thanks to the assumption of an exponential growth homogeneous in the brain). These values are reported in Table 1.

Prion strain 139A ME7 BSE Sc237 RML vCJD Chandler Scrapie 301 V r(day [START_REF] Legname | Continuum of prion protein structures enciphers a multitude of prion isolate-specified phenotypes[END_REF][START_REF] Zampieri | Investigating the conformational stability of prion strains through a kinetic replication model[END_REF]).

In the next section, we find a relation between r and U by investigating their dependence on parameters. Then we show that this relation allows us to fit the data of Table 1 with an accurate choice of the polymerization exponent ν.

Results

Proving the existence of solutions to the eigenvalue problem [START_REF] Greer | A mathematical analysis of the dynamics of prion proliferation[END_REF] is not easy and the theory was first developed in [START_REF] Perthame | Exponential decay for the fragmentation or cell-division equation[END_REF][START_REF] Michel | Existence of a solution to the cell division eigenproblem[END_REF]. In [START_REF] Jauffret | Eigenelements of a general aggregation-fragmentation model[END_REF], we could treat singular polymerization and fragmentation rates as investigated here. It is proved in [START_REF] Jauffret | Eigenelements of a general aggregation-fragmentation model[END_REF][START_REF] Michel | Existence of a solution to the cell division eigenproblem[END_REF] that solutions exist under the condition

γ + 1 -ν > 0. ( 13
)
Moreover, following the idea of [START_REF] Escobedo | On self-similarity and stationary problem for fragmentation and coagulation models[END_REF], we are able to give the dependence of r and U on the parameters V, µ 0 , β 0 and τ 0 for given γ, ν and κ 0 (unknown but strain independent). Define r 1 and U 1 by the eigenequation given for β 0 = V τ 0 = 1 and µ 0 = 0 by

r 1 U 1 (x) = - ∂ ∂x x ν U 1 (x) -x γ U 1 (x) + 2 ∞ x y γ κ 0 x y U 1 (y) dy y . ( 14
)
Then r 1 and U 1 depend only on γ, ν and κ 0 , and we have the following lemma, which we use later on.

Lemma 1 The eigenelements r and U solution of (10) satisfy

r = V τ 0 γ 1+γ-ν β 0 1-ν 1+γ-ν r 1 -µ 0 and U (x) = β 0 V τ 0 1 1+γ-ν U 1 β 0 V τ 0 1 1+γ-ν x . (15)
Proof. The proof is based on a homogeneity argument. First we write [START_REF] Greer | A mathematical analysis of the dynamics of prion proliferation[END_REF] as

r + µ 0 V τ 0 U (x) = - ∂ ∂x x ν U (x) - β 0 V τ 0 x γ U (x) + 2 β 0 V τ 0 ∞ x y γ κ 0 x y U (y) dy y (16)
and we set α = β 0 V τ 0 and R = r+µ 0 V τ 0 . The rescaled function v defined by U (x) = α k v(α k x) satisfies the equation

α k Rv(x) = -α k(2-ν) ∂ ∂x x ν v(x) -α k+1-kγ x γ v(x) + 2α k+1-kγ ∞ x y γ κ 0 x y v(y) dy y . (17)
We choose k = 1 1+γ-ν (which is possible since ( 13) is satisfied) and obtain

α ν-1 1+γ-ν Rv(x) = - ∂ ∂x x ν v(x) -x γ v(x) + 2 ∞ x y γ κ 0 x y v(y) dy y . (18)
Comparing ( 14) and ( 18), the uniqueness of eigenelements ensures that

R = α 1-ν 1+γ-ν r 1 and v(x) = U 1 (x) (19)
because v(x)dx = U (x)dx, and the lemma is proved.

We are now able to give the dependence of r and x on the parameters β 0 , τ 0 and µ 0 . The mean length of the aggregates at time t is defined as the quotient xu(x, t) dx u(x, t) dx . So, using the approximation [START_REF] Greer | Effects of general incidence and polymer joining on nucleated polymerization in a model of prion proliferation[END_REF], we obtain that this quantity is constant in time and equal to

x = x U (x) dx. (20)
Then, thanks to Lemma 1, we conclude the following corollary.

Corollary 1 Under assumptions (7)-( 9) and (13), there are two constants r 1 and x1 (depending only on γ, ν and κ 0 ) such that

r = V τ 0 γ 1+γ-ν β 0 1-ν 1+γ-ν r 1 -µ 0 and x = V τ 0 β 0 1 1+γ-ν x1 . (21)
Proof. Using Lemma 1, we immediately have x = V τ 0 β 0

1 1+γ-ν x1 with x1 = x U 1 (x) dx.
We have obtained the dependence of r and x on the a priori strain-dependent parameters β 0 and τ 0 . Using assumption [START_REF] Laurençot | Well-posedness for a model of prion proliferation dynamics[END_REF], we deduce the dependence of the stability against denaturation

G G = a V τ 0 β 0 1 1+γ-ν x1 + b. (22)
In [START_REF] Zampieri | Investigating the conformational stability of prion strains through a kinetic replication model[END_REF], the exponent γ of the fragmentation rate is equal to 1. Here we do not impose any value for γ but, nevertheless, we consider that it is positive. Indeed the larger the polymers are, the more easily they break themselves. Consequently, a strain-dependency of τ 0 cannot lead to a decreasing correspondence between r and G, as observed in Figure 1, since we have assumed that γ + 1 -ν and a are positive. Thus it indicates that the key strain-dependent parameter should be β 0 . With this assumption, we obtain an implicit expression of G as a function of r :

G = A(r + µ 0 ) 1 ν-1 + b (23)
where A and b are constants (namely strain independent). This model is used to fit the data. First we assume as in [START_REF] Zampieri | Investigating the conformational stability of prion strains through a kinetic replication model[END_REF] that the death rate µ 0 is equal to zero. Then we obtain a model already considered by [START_REF] Zampieri | Investigating the conformational stability of prion strains through a kinetic replication model[END_REF] (Table 3, page 5). In our framework, we can also let µ 0 be free and fit the data with the model [START_REF] Perasso | Identifiability analysis of an epidemiological model in a structured population[END_REF]. As we can see in Table 2, the accuracy of the fitting is a little bit better with this generalization than in the case µ 0 = 0. But the most significant difference between the two fittings is the shape of τ : it is a decreasing function of x in the first case (ν < 0) and it is an increasing function in the second case (ν > 0).

Conclusion and Perspectives

Thanks to a larger class of parameters than [START_REF] Zampieri | Investigating the conformational stability of prion strains through a kinetic replication model[END_REF], we are able to fit experimental data by considering only the fragmentation rate to be dependent on strains. We even obtain a more general model and so a more accurate fitting. But, because the parameter profiles are not a priori known, we cannot find x from experimental data and have to postulate a particular dependence [START_REF] Laurençot | Well-posedness for a model of prion proliferation dynamics[END_REF], whereas this was a consequence of the model in [START_REF] Zampieri | Investigating the conformational stability of prion strains through a kinetic replication model[END_REF]. To validate it, new data are necessary, for instance values of x obtained by an experimental size distribution.

A consequence of the method is the prescription of the shape of the polymerization rate from experimental data, and also the death rate µ 0 . Nevertheless, we notice a significant difference between the two recovered exponents ν obtained by assuming that µ 0 = 0 or not. In particular, one is negative and the other one positive (see Table 2). The idea of comparing parameters from different strains has been deeply exploited here. But the fact that there are only a few different strains in comparison to the number of fitting parameters is a weakness for precise fitting and precise determination of τ. Some parameters such as µ 0 or b should be previously determined by another method and from other data. Furthermore, the parameters of model ( 1) could have a size dependence more general than that given by power laws. A complete inverse problem is to recover all the size-dependent parameters from experimental data without the restriction to a class of shape. This problem is very complicated, even though first elementary models are treated in [START_REF] Doumic | Numerical solution of an inverse problem in sizestructured population dynamics[END_REF][START_REF] Perthame | On the inverse problem for a size-structured population model[END_REF][START_REF] Perasso | Identifiability analysis of an epidemiological model in a structured population[END_REF]. Moreover, it requires more information such as the size distribution U and not only the mean size x.

Aknowledgment

This work has been partly supported by ANR grant project TOPPAZ.

Figure 1 : 1 ν- 1

 111 Figure 1: Experimental values of G and r for different strains are plotted (circles). Then these data are fitted using the reduced model (23) with µ 0 = 0 and µ 0 free.

Table 1 :

 1 Estimated empirical values of the growth rate and the stability against denaturation for different prion strains (data from

	-1 )	0.05 0.024 0.015 0.11	0.18	0.07	0.17	0.07
	G(M )	2	2.9	2.8	1.6	1.7	1.85	2.2	2.2

Table 2 :

 2 Best-fitting parameters obtained with the two different models. The correlation coefficient R 2 is also reported in the last row.

	1 ν-1 +b