Discounted Utility and Present. Value A Close Relation

Transcription

1 Discounted Utility and Present Value A Close Relation Han Bleichrodt a, Umut Keskin b, Kirsten I.M. Rohde b, Vitalie Spinu c, & Peter P. Wakker b a : As b ; corresponding author b : Erasmus School of Economics, Erasmus University Rotterdam, P.O. Box 1738, Rotterdam, 3000 DR, the Netherlands; +31-(0) (F); Wakker@ese.eur.nl; c : The Anderson School at UCLA, 110 Westwood Plaza D511, Los Angeles, CA July, 2013 Abstract We introduce a new type of preference conditions for intertemporal choice, requiring independence of present values from various other variables. The new conditions are more concise and more transparent than traditional ones. They are natural, and directly related to applications, because present values are widely used in intertemporal choice. Our conditions give more general behavioral foundations, 1

2 which facilitates normative debates and empirical tests of time inconsistencies and related phenomena. Like other preference conditions, our conditions can be tested qualitatively. Unlike other preference conditions, however, our conditions can also be directly tested quantitatively, e.g. to verify the required independence of present values from predictors in regressions. We show how similar types of preference conditions, imposing independence conditions between directly observable quantities, can be developed for other decision contexts than intertemporal choice, and can simplify behavioral foundations there Keywords: present value, discounted utility, time inconsistency, preference foundation JEL-classification: D90, D03, C60 2

3 4 1 Introduction Debates about appropriate models of intertemporal choice, both for making rational decisions and for describing and predicting human decisions, usually focus on the critical preference conditions that distinguish between those models. The two most discussed conditions are time consistency, distinguishing between constant and hyperbolic discounting, and intertemporal separability, pertaining to habit formation, addiction, and sequencing effects. 1 Both these conditions are assumed in the classical models but are usually violated empirically. Their normative status has also been questioned. 2 We introduce a new kind of preference conditions, stated directly in terms of present values (PVs). PVs are simple and tractable, and have been widely used in intertemporal choice throughout history (de Wit 1671; Fisher 1930). Hence preference conditions in terms of PVs are simple and natural too. In general, PVs can depend on many variables, such as the times of the receipt of future payoffs, the initial wealth levels at those times, and the wealth levels 1 See Attema (2012), Dolan & Kahneman (2008 p. 228), Epper, Fehr-Duda, & Bruhin (2011), Frederick, Loewenstein, & O Donoghue (2002), Gilboa (1989), Herrnstein (1961), Karni & Safra (1990), Loewenstein & Prelec (1993), Lucas & Stokey (1984 p. 169), Rozen (2010), Strzalecki (2013), Tsuchiya & Dolan (2005), Wakai (2008), and others. 2 See Broome (1991), Gold et al. (1996 p. 100), Parfit (1984 Ch. 14), Strotz (1956 p. 178), and others. 3

4 at other times. Our preference conditions will impose independence of PVs from part of those other variables, and we show that these independencies characterize various models of intertemporal choice. Like all preference conditions, our conditions can be tested qualitatively. Unlike other preference conditions, our conditions can also be directly tested quantitatively. We can carry out regressions with PV as the dependent variable, and the other relevant variables as predictors. 3 We can then test which of those other variables are significantly associated with PV, and whether the variables claimed to be independent in our conditions really are so. Such tests are more widely known and better understood than qualitative tests of preference conditions. They use econometric techniques based on wellestablished probabilistic error theories. To illustrate our new PV conditions clearly, we will apply them to wellknown models. A concise presentation of these models and their representations is in Table 1. Details of the table will be explained in the following sections. The table is presented here because it illustrates the organization of the first four sections. We provide the most concise preference foundations presently available in 3 Predictors are often called independent variables in regressions. The latter term independence does not refer to statistical independence, the concept relevant to our paper. We do not use it so as to avoid confusion. 4

5 the literature for: (a) constant discounted value as commonly used in financial economics (Hull 2013); (b) constant discounted utility (Samuelson 1937); (c) general discounted utility, which includes hyperbolic discounting. We also provide some results that are relevant to other than intertemporal contexts: (d) no-book and no-arbitrage for uncertainty; (e) Gorman (1968)-Debreu (1960) additive separability for general aggregations. All our preference conditions are given in the PV condition column in Table 1. Their statements take little space in this column, but yet give complete definitions as explained later. Further extensions to other decision contexts are a topic for future research, as are extensions to more general intertemporal models (Sections 5 and 7) Preferences and subjective PVs We consider preferences over outcome sequences x = (x 0,..., x T ) R T +1, yielding outcome x t in period t; t = 0 denotes the present. To avoid trivialities, we assume T 2; T is fixed. Besides indexed Roman letters x t that specify the period t of receipt, Greek letters α, β,... are used to designate outcomes (real numbers) when no period of receipt needs to be specified. By α t x we denote x with x t replaced by α. V represents if V : R T +1 R and x y V (x) V (y), for 5

6 Theory Functional Pref. conditions PV condition PV formula T t=0 x t Additive + π(ϕ) ϕ discounted value Symmetric Constant T t=0 λt xt Additive + π(ϕ, t) = τ(ϕ, t + 1) λ t ϕ discounted value Stationary Time-dependent T t=0 λ txt Additive π(ϕ, t) λtϕ discounted value Non- Non- T t=0 U(x t) Separable + π(ϕ, x0, xt) U 1 (U(et + ϕ) discounted utility Symmetric U(et) + U(e0)) e0 Constant T t=0 λt U(xt) Separable + π(ϕ, t, e0, et) = U 1 (λ t U(et + ϕ) discounted utility Stationary τ(ϕ, t + 1, e 1 = e0, e t+1 = et) λ t U(et) + U(e0)) e0 Time-dependent T t=0 U t(xt) Separable π(ϕ, t, x0, xt) U 0 1 (Ut(et + ϕ) discounted utility Ut(et) + U0(e0)) e0 Table 1: The column Functional gives the functional forms. Here: 0 < λ is a discount factor; 0 < λt is a period-dependent discount factor with λ0 = 1; U is a strictly increasing continuous utility function; Ut is a perioddependent strictly increasing continuous utility function. The column Pref. conditions gives preference conditions traditionally used in preference foundations of the functional forms, and defined in the Appendix. The column PV condition gives the PV condition (how π(ϕ, t, e) can be rewritten, with an extra equality imposed for constant discounting) used in our preference foundations. Note that these cells contain complete definitions. The column PV formula gives the formula of PV under each theory. 6

7 56 all x, y R T +1. If a representing V exists, then is a weak order; i.e., 57 is complete (x y or y x for all x, y) and transitive. We assume throughout that is a weak order. Strict preference, indifference, reversed preference, and strict reversed preference ( ) are as usual. We also assume monotonicity (strictly improving an outcome strictly improves the outcome sequence) and continuity of throughout. The following condition considers sums of outcomes x t = ϕ + e t. Here we call e t an initial endowment. We sometimes use the term payoff instead of outcome for ϕ to facilitate interpretations. Similarly, the distinction of the initial endowment only serves to facilitate interpretations and does not refer to any formal reference dependence. Formally, our analysis is entirely in terms of final wealth and is classical in this respect DEFINITION 1 π is the present value (PV ) of outcome ϕ received in period t with (prior) endowment e = (e 0,..., e T ) if (e 0 + π) 0 e (e t + ϕ) t e, i.e. (e 0 + π, e 1,..., e t,..., e T ) (e 0,..., e t + ϕ,..., e T ). (1) Eq. 1 means that, with e the current endowment, receiving an additional future payoff ϕ in period t is exactly offset by receiving an additional present outcome π. That is, the PV of a future payoff ϕ in period t is π. For 7

8 simplicity, we assume throughout that a PV always exists. By monotonicity, it is unique. The definition of PV (denoted π) can be naturally extended to an outcome sequence x with endowment e: (e 0 + π) 0 e e + x. However, we will not need this concept in this paper. The PV can in general depend on ϕ, t, and e, and is a function π(ϕ, t, e). For t = 0, it trivially follows that π(ϕ, 0, e) = ϕ. Note that π is subjective, depending on and, thus, reflecting the tastes and attitudes of the decision maker. The preference conditions presented in the following sections all amount to independence of PV from some of the variables (ϕ, t, e). We denote this independence by writing only the arguments that π depends on. For example, if π(ϕ, t, e) depends only on ϕ (i.e. is independent of t and e) then we write π = π(ϕ). (2) 86 Similarly, if π depends on e only through e 0 and e t, then we write π = π(ϕ, t, e 0, e t ). (3) These conditions, like all other PV conditions in this paper, are indeed preference conditions. Our conditions impose restrictions on indifferences. Preference conditions should be directly verifiable from preferences, the observable primitives in the revealed preference paradigm, without invoking theoretical constructs such as utilities. In Eq. 11 in the appendix we show that our PV 8

9 conditions can be written directly in terms of preferences. They, therefore, are real preference conditions, and our conditions can be tested in the same way as all qualitative preference conditions can be Linear Utility This section considers models with linear utility, as commonly used in financial markets. They can serve as approximations for subjective individual choices if the stakes are small (Epper, Fehr-Duda, & Bruhin 2011; Luce 2000 p. 86; Pigou 1920 p. 785). The first model in Table 1 maximizes the sum of outcomes: T Non Discounted Value: x t. (4) This model does not involve subjective parameters and, hence, is directly observable. It serves well as a first illustration of the nature of PV conditions. t=0 103 THEOREM 2 The following two statements are equivalent: 104 (i) Non-discounted value holds. 105 (ii) π = π(ϕ). 106 Throughout this paper, Condition n.(ii) refers to the condition in State- 107 ment (ii) of Theorem n. Theorem 2 shows that if π depends only on ϕ 108 (Condition 2.(ii)), then it must be the identity function (π(ϕ) = ϕ). This is 9

10 also displayed in the last two columns of the corresponding row of Table 1. It may be surprising that Condition 2.(ii) implies the addition operation of non-discounted value. An informal proof in the appendix will clarify. Our next model involves a subjective parameter, the discount weight λ: Constant Discounted Value: T λ t x t for λ > 0. (5) t=0 113 Under the usual assumption that the decision maker is impatient, we have 114 λ 1. In PV calculations of cash flows, financial economists commonly use constant discounted value, setting λ = 1/(1 + r) with r the interest or discount rate. However, the discount factor λ is then no longer a subjective parameter reflecting the attitude of a decision maker, as in our approach, but it is a given constant, publicly known and determined by the market. The latter approach can be related to our approach if we interpret the market as a decision maker, setting rational PVs, with λ reflecting the market attitude. Condition 2.(ii) rationalizes this common evaluation system. If the parameter λ reflects the attitude of an individual decision maker, then it will probably be influenced by the market interest rate but need not be identical to it. As a preparation for the preference foundation, we define tomorrow s value as an analogue to PV: 126 DEFINITION 3 τ is tomorrow s value of payoff ϕ received in period t (t 1) 10

11 127 with endowment e, denoted τ = τ(ϕ, t, e), if (e 1 + τ) 1 e (e t + ϕ) t e, i.e., (e 0, e 1 + τ, e 2,..., e t,..., e T ) (e 0, e 1,..., e t + ϕ,..., e T ). (6) Now τ is the extra payoff in period 1 that exactly offsets the extra payoff ϕ in period t. It is tomorrow s PV. 131 THEOREM 4 The following two statements are equivalent: 132 (i) Constant discounted value holds. 133 (ii) π = π(ϕ, t) = τ(ϕ, t + 1) = τ Uniqueness of the parameter λ, and all other uniqueness results of this paper, are given in the proofs in the Appendix. Condition 4.(ii) entails that π and τ are independent of the endowments, and that tomorrow s perception of future income is the same as today s. The condition implies that PV depends only on stopwatch time (time differences) and not on calendar time (absolute time). 4 The notation here is short for: π = π(ϕ, t, e) = π(ϕ, t) = τ(ϕ, t + 1) = τ(ϕ, t + 1, e ). The two endowments e and e are immaterial, and are allowed to be different. Because of the shift by one period, the condition is imposed only for all t < T. Existence of π and the equality in Statement 4.(ii) imply that τ also exists. 11

12 Future present values are central tools in recursive intertemporal models (see Caplin & Leahy 2006; Maccheroni, Marinacci, & Rustichini 2006; Ju & Miao 2012 and their references). Stationarity, which requires these future values to be invariant over time, is a convenient special case. Statement 4(ii) formulates this condition in a simplified manner, with only one future payoff and one present value (today or tomorrow), rather than involving general preferences between general outcome sequences as do common versions of stationarity. Most tests of stationarity in the literature are in fact tests of our simplified version; see Takeuchi (2010) and his extensive survey. Many studies have shown that constant discounting is violated empirically. Hence the following model is of interest. Time-Dependent-Discounted Value: T λ t x t (with λ 0 = 1). (7) t= This model allows for general discount weights whose dependence on time is unrestricted. Many special cases of such discount weights have been studied in the literature, the best-known being hyperbolic weighting. The representation in Eq. 7 is not affected if all λ t s are multiplied by the same positive factor. The scaling λ 0 = 1 reflects the most common convention in the literature. 157 THEOREM 5 The following two statements are equivalent: 158 (i) Time-dependent discounted value holds. 12

13 159 (ii) π = π(ϕ, t) An implication, displayed in the last two columns of Table 1, is that if π is any function of ϕ and t, then it must be the function λ t ϕ Nonlinear Utility The models presented in the preceding section take a sum, weighted or unweighted, of the outcomes. They assume constant marginal utility in the sense that an extra euro received in a particular period gives the same utility increment regardless of the endowment of that period. In individual choice this condition is often empirically violated and not normatively imperative. More realistic and more popular models allow for nonlinear utility. If marginal utility does depend on the endowment, then the models of the preceding section become: T Non-Discounted Utility: U(x t ); (8) Constant Discounted Utility: Time-Dependent Discounted Utility: t=0 T λ t U(x t ); (9) t=0 T U t (x t ). (10) t= Eq. 9 is Samuelson s (1937) discounted utility, the most popular model for intertemporal choice. Each utility-model reduces to the corresponding value- 13

14 model if utility is taken linear. In particular, in time-dependent discounted utility, if the U t (α) are linear then they can be written as λ t α, where we can renormalize such that λ 0 = 1. Then time-dependent discounted value results. The mathematics underlying the preference foundations for the utility models in Eqs is more advanced than it was for Theorems 2, 4, and 5. Whereas those theorems solved linear equalities, we now have to deal with general equalities, with nonlinear utilities intervening. Fortunately, the increased complexity of the underlying mathematics does not show up in the preference conditions and, consequently, in the empirical tests of the models. The relevant PV preference conditions are obtained directly from those defined in 3 by adding dependence on the endowment levels e 0, e t. This way we readily obtain Theorems 6, 7, and 8 from Theorems 2, 4, and 5, respectively. The transparancy of our preference conditions clarifies the logical relations between the different results. 188 THEOREM 6 The following two statements are equivalent: 189 (i) Non-discounted utility holds. 190 (ii) π = π(ϕ, e 0, e t ). 5 5 Here t refers to the period where ϕ is added and e t specifies the endowment in that period. Note that there is no explicit dependence of π on t, which we denote by suppressing 14

15 An implication, displayed in the last two columns of Table 1, is that if π is any function of ϕ, x 0, and x t, then it must be of the form displayed there. Several authors have argued that any discounting, even if consistent over time, is irrational, and have thus recommended non-discounted utility for intertemporal choice (Jevons 1871 pp ; Ramsey 1928; Rawls 1971; Sidgwick 1907). Condition 6.(ii) characterizes this proposal. Studies providing preference foundations for non-discounted utility (sums of utilities) include Fishburn (1992), Kopylov (2010), Krantz et al. 1971), Marinacci (1998), Pivato (2013), and Wakker (1989). Blackorby, Davidson, & Donaldson (1977) provided results for lotteries with T + 1 equally likely lotteries, taking the representation as von Neumann-Morgenstern expected utility. We now turn to discounting. 203 THEOREM 7 The following two statements are equivalent: 204 (i) Constant discounted utility holds. 205 (ii) π = π(ϕ, t, e 0, e t ) = τ(ϕ, t + 1, e 1 = e 0, e t+1 = e t ). 6 period t from the arguments of π(...). 6 The condition implies that the endowment levels e j, j 0, t of the PV endowment e, and similarly the endowment levels e j, j 1, t + 1 of the tomorrow-value endowment e, are immaterial. Existence of π and the equality imply that τ also exists. The condition holds for all 0 t T 1, and whenever e 1 = e 0 and e t+1 = e t. We use the convenient argument matching notation popular in programming languages (R: see R Core Team 2013, 15

16 The first preference foundation of constant discounted utility was in Koop- mans (1960, 1972), with generalizations in Harvey (1995), Bleichrodt, Ro- hde, & Wakker (2008), and Kopylov (2010). Condition 7.(ii) requires that 209 the same tradeoffs are made tomorrow as today. Such requirements have 210 sometimes been taken as rationality requirements (see the introduction). 211 THEOREM 8 The following two statements are equivalent: 212 (i) Time-dependent discounted utility holds. 213 (ii) π = π(ϕ, t, e 0, e t ) Statement 8.(ii) expresses that the tradeoffs between periods 0 and t are independent of what happens in the other periods, reflecting a kind of separa- bility. Time-dependent discounted utility is a general additive representation, 217 which has been axiomatized several times before. 7 It implies intertemporal separability, which is arguably the most questionable assumption of most intertemporal choice models. Python: see Python Software Foundation 2013, Scala: see École Polytechnique Fédérale de Lausanne 2013, among others) to express the latter two restrictions. Here the formal argument of a function (e 1 or e t+1) is assigned a value (e 0 or e t ). 7 Debreu (1960), Gorman (1968), Krantz et al. (1971); Wakker 1989). 16

17 Applications to contexts other than intertem- poral choice The mathematical theorems provided in previous sections, and the preference conditions that we used, can be used in contexts other than intertemporal choice. For instance, Theorem 5 is of special interest for decision under un- certainty, capturing nonarbitrage in finance. To see this point, we reinterpret the periods t as states of nature. Exactly one state obtains, but it is uncer- tain which one (Savage 1954). Now x = (x 0,..., x T ) refers to an uncertain prospect yielding outcomes (payoffs) x t if state of nature t obtains. For sim- plicity, we focus on a single time of payoff here, so that discounting plays no role. If we divide the discount weights λ t by their sum T t=0 λ t (relaxing the requirement of λ 0 = 1), then they sum to 1, and the representation becomes subjective expected value. Subjective expected value was first axiomatized by de Finetti (1937) using a no-book argument, called no-arbitrage in finance. Then the representation is as-if risk neutral, and the decision maker is the market which sets rational prices for state-contingent assets. For state j, a state-contingent asset x = (0,..., 0, 1, 0,..., 0) yields outcome 1 if j happens and nothing otherwise. In this interpretation, λ j becomes the market price of this state-contingent asset, 239 and PV is the offsetting quantity of state-0-contingent assets. Condition 17

18 (ii) provides the most concise formulation of the no-book and no-arbitrage argument presently available in the literature. For decision under uncertainty, certainty equivalents are more natural 243 quantities than state-contingent prices. For risk, Chew (1983) explained basic techniques for applying functional equations to certainty equivalents to obtain preference foundations. Eq. 8 can be interpreted as von Neumann-Morgenstern expected utility for equal-probability lotteries, which essentially covers all lotteries with rational probabilities (writing every probability i/j as i probabilities 1/j). Eq. 8 can also be interpreted as ambiguity under complete absence of information (Gravel, Marchant, & Sen 2012). Eq. 10 concerns Debreu s (1960) additively separable utility. Here again, our Statement 8.(ii) provides the most concise preference axiomatization presently available in the literature Discussion The primary purpose of preference foundations is not to provide surprising logical relations with deep and complex proofs, as it would be in abstract mathematics. Their primary purpose is to make decision models with theoretical constructs directly observable, by restating their existence (Statements (i) in our theorems) in terms of the preference relation (Statements 18

19 (ii) in our theorems). The simpler the preference conditions, the better they clarify the empirical meaning of the decision models. Similarity between the preference conditions and the functional is an advantage, because it helps to clarify the empirical meaning of the decision model. Hence the major motivation for this paper has been to introduce preference foundations that are as simple as possible and that reflect the corresponding decision models as well and transparently as possible. The conditions in our Statements (ii) use fewer words and characters than any conditions previously proposed in the literature, and thus are the most concise ones presently existing. The surprising aspect of our paper is that conditions for preference foundations of intertemporal decision models can be that simple. Further, our conditions are easy to understand and test because present values are familiar objects. Our present value conditions use indifferences rather than preferences. Conditions for indifferences are logically weaker, making their implications logically stronger. 8 Our conditions only concern the simplest tradeoffs possible involving the change of the present payoff and one future payoff, further enhancing their generality. Thus commonly used stationarity conditions are more restrictive than the conditions in our Statements 6(ii) and 7(ii). 8 They follow from preference conditions by applying the latter twice, first with preferences one way and then with preferences reversed. 19

20 An empirical advantage of our preference conditions is that they can be directly tested using analyses of variance and regression techniques. If we take PV as the dependent variable, then Eq. 3 predicts that ϕ, t, e 0, and 281 e 1 may be significant predictors, but not the e j s with j 0, t. We can test this prediction using common regression analyses, which allow us to use the probabilistic error theories underlying econometric regressions, which are easier to use than the more recently developed error theories for preferences (Wilcox 2008). The follow-up paper Keskin (2013) provides extensions of our results to some popular hyperbolic discount models, still maintaining intertemporal separability. Generalizations of the latter condition to situations with partial separability 9 are a topic for future research Conclusion We have introduced a new kind of preference conditions for intertemporal choice, requiring that (quantitative) PVs are independent of particular other variables. The resulting conditions can be used both in qualitative and in 9 Gilboa (1989)and Zuber & Asheim (2012) maintain rank-dependent separability. Le Van, Cuong & Yiannis Vailakis (2005) maintain Bellman s equation. Noor (2009) gives up time-independence of utility. 20

21 294 quantitative tests, and are more concise and transparent than preceding con- 295 ditions proposed in the literature. The technique of obtaining preference conditions by taking a quantity with direct empirical meaning and then imposing independence conditions can be extended to other decision contexts such as decision under uncertainty. Then new, more concise characterizations result, which can easily be tested empirically. 21

22 Appendix. Proofs, uniqueness results, and clarification of the empirical status of PV conditions We present the proofs, with added the uniqueness results, of our theorems in reversed order, from most to least general, because this approach is most clarifying and most effi cient. The presentation of the proofs clarifies the relationship between our PV conditions and the well-known preference conditions, showing that PV conditions indeed are preference conditions. In each proof, we start from our PV condition, which is always weaker than the conditions that are derived and that are commonly used in the literature. We thus show that our PV conditions give stronger results. Because each Statement (ii) is immediately implied by substitution of the functional, we throughout assume Statement (ii) and derive Statement (i) and the uniqueness results P T 8, The following uniqueness result holds for Statement 8.(i), which uses Eq. 10: A time-dependent real constant µ t can be added to every U t, and all U t can be multiplied by a joint positive constant ν. This follows from well- 22

23 known uniqueness results in the literature (Krantz et al Theorem 6.13; Wakker 1989 Observation III.6.6 ). We next derive Statement 8.(i). The equality π(ϕ, t, e) = π(ϕ, t, e 0, e t ) in Condition 8.(ii) means that π is independent of e j, j 0, t. This holds if and only if (e 0 + π, e 1,..., e t 1, e t, e t+1,... e T ) (e 0, e 1,..., e t 1, e t + ϕ, e t+1,... e T ) (e 0 + π, e 1,..., e t 1, e t, e t+1,... e T ) (e 0, e 1,..., e t 1, e t + ϕ, e t+1,... e T ).(11) 322 This holds if and only if the implication holds with twice preference instead 323 of indifference. 10 Eq. 11 with preference instead of indifference is known as separability of {0, t} (Gorman 1968). By repeated application of Gorman (1968), separability of every set {0, t} holds if and only if is separable; i.e., every subset of {0,..., T } is separable (preferences are independent of the levels where outcomes outside this subset are kept fixed, as with separability of {0, t}). This holds if and only if an additively decomposable representation holds 11, which we call time-dependent discounted utility in the main text To wit, if the upper indifference is changed into a strict preference, then we find π < π to give indifference. The lower indifference follows with π instead of π. By monotonicity, replacing π by π leads to the lower strict preference. 11 See Debreu (1960), Gorman (1968), Krantz et al. (1971 Theorem 6.13), Wakker (1989 Theorem III.6.6). 23

24 P T 7, The following uniqueness result holds for Statement 7.(i), which uses Eq. 9: A real-valued time-independent constant µ can be added to U, and U can be multiplied by a positive constant ν. λ is uniquely determined. These uniqueness results follow from those in Theorem 8, where, in terms of Eq. 10, λ is the proportion U t+1 /U t for any t. It is useful to note that the sum of weights, λ t, is the same for each outcome sequence evaluated, implying that there is no special role for utility value 0. We next derive Statement 7.(i). The equality π(ϕ, t, e 0, e t ) = τ(ϕ, t + 1, e 1 = e 0, e t+1 = e t ) in Condition 7.(ii) means that π is independent of e j, j 0, t, T (as in Theorem 8, but now only for t < T ), but also of whether it is measured in period 0 or period 1. This holds if and only if, writing α for e 0 = e 1 and β for e t = e t+1, (α + π, e 1,..., e t 1, β, e t+1,... e T ) (α, e 1,..., e t 1, β + ϕ, e t+1,... e T ) (e 0, α + π,..., e t 1, e t, β,... e T ) (e 0, α,..., e t 1, e t, β + ϕ,... e T ).(12) It implies separability of {0, t} for all t < T, as in Theorem 8, but now, instead of separability of {0, T } we have separability of all {1, t + 1} for all t < T. The latter separability can for instance be seen by replacing all primes in Eq. 12 by double primes, which should not affect the second indifference because of the maintained equivalence with the first indifference. By repeated 24