The aggregation of preferences: can we ignore the past?

Transcription

1 The aggregation of preferences: can we ignore the past? Stéphane Zuber Toulouse School of Economics Abstract The present paper shows that a Paretian social objective can be history independent and time-consistent only if a stringent set of conditions is verified. Individual utilities must be additive. The social welfare function must be a Utilitarian aggregation of those utilities. If we moreover want social preferences to be stationary, all individuals must have the same discount rate. The results, obtained in a general setting, are implemented in several frameworks: deterministic dynamic choice; stochastic dynamic choice, with von Neumann and Morgenstern utilities, and then with Kreps and Porteus preferences. These applications highlight the restrictiveness of the conditions uncovered. Keywords Social welfare function, aggregation of preferences, history independence. JEL Classification Numbers D69, D71, D99. I wish to thank Antoine Bommier for many valuable discussions and comments. Toulouse School of Economics, Université Toulouse 1, Manufacture des Tabacs, 21 allée de Brienne, Toulouse, France. Phone Number: +33 (0) Stephane.Zuber@univ-tlse1.fr

2 1 Introduction Dynamic welfare economics aims at providing social evaluation criteria and policy guidance for intertemporal issues. It stands at the intersection of two branches of economic theory: welfare economics and dynamic choice theory. Not surprisingly, the literature has focused on criteria satisfying properties coming from those two fields. In this respect, the Pareto axiom and the property history independence are generally considered as mild requirements. Most people maintain that any valid social aggregation of preferences should respect individuals choices. This principle is represented by the well-known Pareto axiom, the cornerstone of welfare economics and of modern individualistic ethics. The majority of intertemporal social evaluation criteria suggested in the literature satisfy some version of this principle, whether the horizon is finite (see Blackorby and al., 2002) or infinite (see the literature on the aggregation of infinite utility streams, surveyed for instance in Fleubaey and Michel, 2003). In models of intertemporal decision making, it is however convenient to assume that preferences satisfy specific properties. Quite naturally, those properties have been used in dynamic social choice. For instance, the predominant intertemporal social objective, used in seminal models of optimal growth (Ramsey 1, 1928; Koopmans, 1965) and of optimal resource depletion (Dasgupta and Heal, 1979), is additively separable and exhibits exponential discounting: + t=0 δt U t, where U t is the aggregate welfare at time t and δ is the social discount factor. Subsequent research has postulated more general recursive social objectives (Beals and Koopmans, 1969; Uzawa, 1968), allowing for endogenous discounting. All these objectives remain characterized by the fact that preferences at a given period are independent of what happened in the past: the social welfare function is history independent. Actually, the mentioned objectives are stationary, a stronger 1 Ramsey (1928) actually focuses on the special case δ = 1 and uses a slightly different criterion: W = + t=0 (B U t), where B represents the bliss (the maximal reachable level of welfare). 1

3 property. In this paper, we investigate the conditions under which a Paretian, timeconsistent and history independent aggregation is possible. We find that a major condition bears on possible individual utilities: they must be additively separable. Another condition bears on the form of social aggregation, which must be Utilitarian. We also find necessary and sufficient conditions for obtaining stationarity and Paretian social preferences: individuals preferences must display exponential discounting and people must have the same rate of impatience. Some of these results resemble to those obtained separately by Blackorby, Bossert and Donaldson (2005) in the case of a finite number of overlapping generations. Our approach is different, for we work directly on preferences and within an infinite horizon framework in order to discuss criteria used in the macroeconomic literature. Besides our general description of the intertemporal problem in Section 2 permits to consider a wider range of intertemporal situation, so that we are able to obtain a more general answer to the issue we study. Indeed our findings are obtained in an abstract framework for dynamic choice, designed to encompass individual and social problems in many intertemporal choice settings. The framework and some properties satisfied by intertemporal preferences are introduced in the next section. Section 3 distinguishes individual and social problems and presents social aggregation. The difficulty of aggregating preferences is illustrated by a simple example in Section 4. Then we state the results on possible aggregations in the general framework. We derive and discuss their consequences in specific frameworks: deterministic dynamic choice (Section 5.1), stochastic dynamic choice with von Neumann and Morgenstern preferences (Section 5.2), and with Kreps and Porteus preferences (Section 5.3). Those examples highlight the restrictiveness of the conditions for social aggregation. In addition, the additive forms we find have several implications in terms of inequality aversion and preference for mobility that might seem ethically unappealing. The last section contains some concluding remarks. Proofs of the propositions 2

4 are relegated to an appendix. 2 Dynamic choice Throughout the paper, N is the set of positive integers, R the set of real numbers, and R + the set of non-negative reals. For n N, R n is the Euclidian n-space, and R n + the non-negative orthand of the Euclidian n-space. t indicates time, which is discrete: t N. 2.1 A general setting For each period, there is a set of payoffs X. For the sake of simplicity, and because we want to be able to consider stationary problems, the payoff set is assumed to remain the same as time elapses. A payoff history at time t is y Y t, where Y t is defined recursively by Y t+1 = Y t X and Y 1 = X 0, with X 0 any convenient singleton set. The set of actions that can be undertaken is A, the same for each period. Actions in some way determine present and future payoffs: a A describes prospects of payoffs. Throughout the paper, all sets satisfy desirable topological properties. Namely they constitute complete separable metric space and are connected. We also assume that X A is homeomorphic to a subset of A. This is commonly noted X A A, though this is strictly speaking improper. An obvious example is when actions only involve choosing the infinite sequence of payoffs, as in the deterministic dynamic case. Indeed we then have A = i=1 X = X i=1 X. In stochastic dynamic choice, sequences of payoffs are the random results of actions. But the hypothesis is still verified 2. A dynamic choice problem consists of a sequence of choices on A, given realized past payoffs. We assume that for each t N and each y Y t, choices can be 2 In this case, the set of lotteries in A for which the first-period outcome is deterministic is homeomorphic to X A. 3

5 represented by a continuous total preorder y, t on A. 3 A famous result by Debreu (1954) states that for each history there exists a real-valued order-preserving continuous utility function, U y, t : A R : a, â A, y Y t : a y, t â U y, t (a) U y, t (â) Preferences on X A correspond to the restriction of y, t to this subset of A. We also assume that dynamic preferences are sensitive in all periods in the following sense. There exist x and ˆx in X such that U y,t (x, a) > U y,t (ˆx, a) for all y Y t and all a A; and there exist a and â in A such that U y,t (x, a) > U y,t (x, â) for all y Y t and all x X. 2.2 Properties of temporal preferences It is customary to narrow the scope of possible preferences by assuming that they satisfy particular properties. A widespread assumption is that choices are consistent: the plan selected at time t must also be adopted at time t + 1, taking into account payoff history. This is a key property of rational decision making. Property 1 (Consistency) Preferences are (time-)consistent if, for any t N, y Y t, x X, a and â in A: U y, t (x, a) U y, t (x, â) U (y,x), t+1 (a) U (y,x), t+1 (â) Property 1 is similar to axiom 3.1 in Kreps and Porteus (1978) and to the strict consistency axiom in Streufert (1998). 4 As they show, this implies (under the sensitivity condition) that there exists an aggregator function V y, t (.,.) 3 A total pre-order on a set is a complete, reflexive and transitive relation on that set. The definition of y, t being continuous is as follows: for any a A, the sets {x A : x y, t a} and {x A : a y, t x} are closed under the topology for A. In a conventional way, and are the asymmetric and symmetric part of. 4 The strict consistency axiom of Streufert (1998) combines consistency and history independence. 4

6 that is continuous and strictly increasing in its second argument and such that: ) U y, t (x, a) = V y, t (x, U (y,x), t+1 (a), (x, a) X A. This first property is usually combined with a second one, called history independence: Property 2 (History Independence) Preferences are history independent if, for any t N, y Y t, a and â in A: U y, t (a) U y, t (â) = Uŷ, t (a) Uŷ, t (â) ŷ Y t If preferences are history independent, we can drop the subscript y of the utility functions: U y, t (a) = U t (a). Combined with the consistency axiom, this ) yields the classical recursive formulation: U t (x, a) = V t (x, U t+1 (a), (x, a) X A. Recursive preferences are commonplace in the literature. The reason for this success is that they are a simple way to ensure consistency of temporal choices 5. This is too strong an assumption though, as illustrated by the literature on habit formation, which considers history-dependent choices. But the modelling of how past payoffs affect current preferences can be complicated and involves an elaborate mathematical structure. Unless some particular structure for history dependence is assumed, all past payoffs need to be state variables of the dynamic programming problem. As time elapses, the increasing number of state variables can render the resolution of the problem particularly difficult. The simplicity and practicality of the recursive specification explain its success. A related but stronger assumption is that preferences are stationary. 5 See Blackorby and al. (1973) and Johnsen and Donaldson (1985) who study the situation of a finite time-horizon. 5

7 Property 3 (Stationarity) Preferences are stationary if, for any x X, a and â in A: U 1 (a) U 1 (â) U 1 (x, a) U 1 (x, â) This definition of stationarity is consistent with the definition given, for instance, by Koopmans (1960). Stationarity implies that, t N and (x, a) ( ) X A, U t (x, a) = U(x, a) = V x, U(a). Choices are then clearly consistent and history independent. In addition, preferences on X A do not change with time. Stationarity is a key ingredient of most models of intertemporal planning. In line with the restrictions imposed on the form of utilities by Properties 1 to 3, it is usual to consider that preferences on X A are defined by an aggregator function combining preferences on current payoffs and preferences on prospects ) for the future: U t (x, a) = V t (u t (x), U t (a). Some form of this aggregator function V t (u, U) are of particular interest. First it can be additive: Definition 1 (Additivity) Preferences on X A are (time-)additive if, for any t N, y Y t, x X, a A, they are represented by: ) U y,t (x, a) = V t (u t (x), U t (a) = u t (x) + δ t U t (a) for some positive real numbers δ t. If preferences are time-additive, they are obviously consistent and history independent. Additivity adds the further condition that the prospects for the future do not affect choices between current payoffs. Indeed, additive preferences satisfy the following property of independence of the future: U y, t (x, a) U y, t (ˆx, a) U y, t (x, â) U y, t (ˆx, â) for all â A The stationary counterpart of additivity is exponential discounting: 6

8 Definition 2 (Exponential discounting) Preferences on X A exhibit exponential discounting if, for any t N, y Y t, x X, a A, they are represented by: U y,t (x, a) = V ( u(x), U(a) ) = u(x) + δu(a) for some constant positive real numbers δ. When preferences on X A are represented using an aggregator function ( V V t u t, U t ), the discount factor can be defined as in Koopmans (1960): t U t ut=ut. In the case of exponential discounting, the discount factor and discount rate are constant and respectively equal to δ and ρ = 1 δ. Stationary preferences with δ a constant rate of discount have played a prominent role in the literature ever since Samuelson (1937). Nevertheless, in major contributions, Koopmans (1960), Koopmans, Diamond and Williamson (1964) and Uzawa (1968) have strongly argued in favor of utilities displaying endogenous discounting instead. 3 Aggregating individual preferences The model of choice laid out in Section 2 applies to individuals or to a social planner. Below we discuss the interelation between individuals and the planner s preferences. A social aggregation indeed derives the latter from the former. 3.1 Planner s and individuals preferences Society is composed of a finite (but possibly very large) number N N of infinitely-lived individuals. The set of individuals is denoted I = {1,, N}. We assume that all individuals have the same payoff and action sets for each period: X and A. The individual payoff history set, Y t, is constructed as indicated in Section 2.1. To denote the payoff of a particular individual at a specific date, 7

9 we use the notation x i t (payoff of individual i I at period t N). Similar notation are used for actions, histories and utility or aggregator functions: subscripts always refer to time, and superscripts to individuals. The only hypotheses on individual preferences are that they are self-regarding, well-behaved (transitive, complete and continuous) and sensitive. i is the yt i, t preference ordering on A of individual i I at time t N given his payoff history y i t Y t. U i y i t, t(.) is a continuous utility function representing i y i t, t. Social choice sets are defined as follows. X = i I X, the cartesian product of individual payoff sets, is the social payoff set at any period. Similarly, the social payoff history set at time t is Y t = i I Y t and the social action set is A = i I A, the same each period. A generic element of X at time t is x t, and x i t indicates the ith component of x t. X i refers to the subset of the space of social payoffs corresponding to individual i s payoffs. Similar notation are used for Y t and A. Social preferences are represented by an ordering S y t, t on A. We adopt for the social planner the model of choice of Section 2. This implies that his preferences are continuous. There are multiple examples in social choice theory proving that this assumption is not innocuous. Therefore, we prefer to state the property separately: Assumption (Continuity) Social preferences are continuous. Continuity guarantees that there exists a continuous social welfare function U S y t, t(.) : A R that represents the social preference ordering S y t, t. 3.2 Social aggregation The aim of a social aggregation is to derive the social preference ordering from individual utility functions. Many ethical principles can be invoked to determine how this aggregation should be made. But the most widely accepted principle is that social preferences should be Paretian: 8

10 Axiom (Pareto) Social preferences are Paretian if for any t N, y t Y t, a t and â t in A: a i t i y i t,t âi t i I a t S y t,t â t If furthemore there exists j I such that a j t j y j t,t âj t then a t S y t,t â t Pareto s principle asserts that the unanimous preference of individuals must be respected by the social planner. This principle is the cornerstone of individualistic ethics. Almost all social criteria that have been proposed satisfy a version of this axiom. The Pareto Axiom has an important implication concerning the form of the social welfare function when it is continuous. Indeed, it can be proved that social preferences are continuous and Paretian if at any period t, there exists a social aggregator function U yt, t, continuous and increasing in all its arguments, such that the social welfare function can be expressed (see Fleurbaey and Mongin, 2005) as: ( ) Uy S t, t(a t ) = U yt, t U 1 yt 1, t(a1 t ),, U N yt N, t(an t ) (1) for all a t in A and for all y t in Y t. Pareto s principle is an ordinal notion. We only need to know individual preference orderings, a very weak requirement regarding the informational basis of the social choice. Indeed, if we were to change the utility functions representing individual preferences, we would only have to change the aggregator function to get exactly the same social welfare function. 6 Similarly, we only use ordinal concepts along the paper. It is also fruitful to consider a more specific, additively separable, form of the social aggregator function. The formula is widespread in the literature on the 6 For more on this, and for a recent defense of the Bergson-Samuelson social welfare function, see Fleurbaey and Mongin (2005). 9

11 aggregation of intertemporal preferences (see Blackorby et al., 2002). Besides, it is obtained as a result in our main propositions. We call of this form of aggregation Utilitarian : Definition 3 Social preferences at time t are a Utilitarian aggregation of preferences if there exists a profile of individual utility functions (U y 1 t,t(.),, U y N t,t(.)) such that, for all y t Y t, a t A: U 1 y 1 t,t(a1 t ) + + U N y N t,t(an t ) (2) represents the planner s preferences for a t. One can feel uncomfortable with the name Utilitarianism to describe the additive form in Equation (2). Indeed Utilitarianism conventionally requires cardinal representations of utilities. Definition 3 uses ordinal representations, in order to be consistent with the rest of the paper. Bearing in mind this caveat, the name Utilitarian is used for the sake of simplicity. 4 Possible aggregations We are now able to address the core issue of the paper: under what conditions a consistent Paretian aggregation of preferences can be history independent? As a prelude to answering that question, an example can suggestively illustrate the difficulty of the task. It shows that, even if individuals have the same stationary preferences, the Utilitarian aggregation of their utilities may not be history independent. 10

12 4.1 Example Consider a society of two individuals having the same stationary utilities, suggested in Koopmans et al. (1964): U(x i, a i ) = 1 θ log ( 1 + [u(x i )] ε + δu(a i ) ), (x i, a i ) X A, i {1, 2} with 0 < δ < 1, θ > 0 and ε > 0. Assume that the social welfare function at the initial period is defined by the simple Utilitarian aggregation U S 1 (x, a) = U(x 1, a 1 )+U(x 2, a 2 ), for any x X X and a A A. We obtain: U1 S (x, a) = 1 )( )] [(1 θ log + [u(x 1 )] ε + δu(a 1 ) 1 + [u(x 2 )] ε + δu(a 2 ) We have seen in Section 2.2 that, if the social objective is consistent, it must be expressed U S 1 (x, a) = V S 1 (x, U S x,2(a)). In the present instance, this form is possible if and only if V S 1 (x, U S x,2(a)) = 1 θ log [ 1 + [u(x 1 )] ε + [u(x 2 )] ε + ([u(x 1 )][u(x 2 )]) ε + δu S x,2(a) ] and U S x,2(a) = [u(x 1 )] ε U(a 1 ) + [u(x 2 )] ε U(a 2 ) + U(a 1 ) + U(a 2 ) + δ U(a 1 ) U(a 2 ) Hence Ux,2(a) S is not independent of x. In view of this example, combining Paretianism and history independence appears to be far from a trivial task. The case of the Utilitarian aggregation shows that strong hypothesis on individual preferences are likely to be needed: stationarity - which implies consistency and history independence - being for instance not sufficient. The next section shows that this intuition is correct in the broader context of Paretian aggregations. Note in passing that our example 11

13 also shows that not all stationary preferences can be aggregated into a stationary social objective. 4.2 Possibility results for an amnesic planner We want to find the conditions under which it is possible to aggregate individual preferences into a social objective satisfying history independence and consistency. We require the aggregation be Paretian. Proposition 1 Suppose that the social planner is Paretian. His preferences are consistent and history independent if and only if individuals have additive preferences and social preferences are a Utilitarian aggregation of those additive representations. Proof. See the appendix. The proof makes use of separability theorems by Debreu (1959) and Gorman (1968). Proposition 1 is the main result of the paper. It indicates that Pareto s principle combined with history independence and consistency reduces the possibilities of social choice. We indeed find that restrictions both on individuals utilities and on the form of the social aggregation are imposed. Restrictions on individuals preferences mean that it is not always possible to find social choice rules satisfying Pareto s principle and having the history independent recursive form. Whenever individuals utilities do not meet the conditions stated in Proposition 1, it is impossible to aggregate preferences into a history independent and consistent social objective. Restrictions on the form of the social aggregation mean that the possibilities for the social planner to trade-off individuals welfare are also limited. This may have important ethical consequences. However it is difficult to assess how strong the restrictions are. To do so, we need to consider specific settings. That is why Section 5 discusses the implications of Proposition 1 in various contexts. 12

14 The economic literature frequently makes the stronger assumption that the social objective is stationary. The most common form of stationary utility is exponential discounting. But the example of section 3.3 reminds us that the class of stationary preferences is pretty wide-ranging. In particular, it is possible to find stationary objectives with non-constant discount rates (see Koopmans, 1960; Koopmans and al., 1964). The following proposition indicates that such objectives cannot be the result of a Paretian aggregation: Proposition 2 Suppose that social preferences are Paretian. Then the social objective is stationary if and only if individuals and the planner have preferences exhibiting exponential discounting and have the same rate of time preference. Proof. See the appendix. Proposition 2 makes it clear that a specific form of stationarity is required for individual utilities, namely exponential discounting. This may not be surprising given Proposition 1: exponential discounting corresponds the class of additive preferences which are stationary. More striking is the condition that people must have identical time preferences. It seems that people behave differently in intertemporal choice situations, so that they more probably have very heterogeneous patience. Any departure from the homogeneous patience case however introduces non-stationarity in the planner s objective. 5 Applications Additivity and exponential discounting are the major conditions imposed by the Propositions of Section 4.2. Their definitions in Section 2.2 indicate that they only apply to preferences on X A and X A. Since the properties of consistency, history independence and stationarity were also defined on these restricted 13

15 domain, finding that the conditions bear only on restricted preferences makes sense. But we can show that the propositions also have significant implications for choices on A and A in several frameworks of particular interest. This also permits to better assess the restrictiveness of the conditions in Propositions 1 and Deterministic dynamic choice The simplest and most natural framework dealing with intertemporal preferences is deterministic dynamic choice. This class of problems consider choices over (consumption) programs. A program at time 1 is a a sequence χ i = (x i 1,, x i t, ), with each x i t belonging to X. The most common interpretation is that X is a commodity or consumption space, which is generally represented by a compact subset of R n +. At period t, choices are made over programs for present and future consumption, which are sequences t χ i = (x i t,, x i t+t, ), where xi t X. Clearly, a i t = t χ i, so that the choice set is A = X. Besides, each sequence t χ i can also be written (x i t, t+1 χ i ): X A and A are strictly identical spaces. Actions at time t only involve choosing a sequence of present and future payoffs. Once we have preferences on X A, we can directly deduce the form of preferences on A. In Proposition 1, it was showed that individual i s preferences for any sequence tχ i in A must be represented by a utility function U i t (x i t, t+1 χ i ) = u i t(x i t) + δ t U i t+1( t+1 χ i ) Proceeding by induction yields the additively separable representation of preferences over programs: U i t (x i t,, x i T, ) = β t,τ u i τ(x i τ) (3) τ=t 14

16 with β t,t = T 1 τ=t δ τ. Additively separable forms have the noticeable consequence that trade-offs between goods at each period are independent of what happens in other periods. Preferences are said to be period-independent. This property precludes any complementarities between consumptions at different periods, which is restrictive from a theoretical perspective and clearly absent from empirical data (see Deaton, 1971). Proposition 2 can also be extended to choices on sequences t χ i. Individual preferences must then be represented by the utility function: U i t (x i t,, x i T, ) = δ τ t u i τ(x i τ) (4) τ=t Individuals may have different immediate utility functions, u i t(.), but they necessarily have the same discount rate, ρ = 1 δ. Consequently they must have the δ same way of making trade-offs between present and future advantages. A second drawback of Equation (4) is that this entails assuming constant discounting. As stated by Koopmans, the constant discount rate seems too rigid to describe important aspects of choice over time (Koopmans, 1960, p. 308). That is why case of Koopmans, Diamond and Williamson (1964) proposed stationary preferences with a discount factor decreasing in wealth, which, they argued, may be found more realistic. The form of social aggregation also matters. Proposition 1 states that social welfare at period t is ordinally equivalent to β t,τ u i τ(x i τ) (5) τ=t i I Social welfare is therefore also period-independent. In such a case, the distribution of advantages at each point of time is independent of what people obtained in the past and of their prospects. Each individual life s period is treated sepa- 15

17 rately, which precludes any idea of compensation for intertemporally correlated bad outcomes. In this situation, an indifference toward intertemporal inequalities similar to the one discussed in Atkinson and Bourguignon (1982) arises. Besides, since social welfare is a sum of time-separable utilities, the mobility structure is irrelevant (see Shorrocks, 1978). A society where the poor always remain poor and the rich always remain rich is equivalent to a society where social positions switch, provided there is always the same number of rich and poor. There are few grounds for believing that this feature of social preferences accords with prevailing ethical views. Intertemporal mobility is usually thought to be desirable. Proposition 2 also induces a specific form for social welfare: U S t (x t,, x T, ) = δ τ t u S τ (x τ ) (6) τ=t with u S t (x t ) = i I ui t(x i t). The social planner has preferences exhibiting a constant discount rate, which is the same as the individual one. 5.2 Stochastic dynamic choice: vnm preferences A second conventional reference framework takes into account uncertainty. In that case, X is the one-period consumption set. An infinite consumption stream belonging to X describes consumptions for all periods in the future. It is similar to programs studied above. Choices are made among stochastic or uncertain consumption streams. Thus, we take A to be the set of Borel probability measures on the measurable space (X, R(X )), with R(X ) the Borel σ-field of X. Assuming that X is a compact subset of R +, A is a compact separable metric space. Social sets are the product sets of individual ones. 7 Choices on X A can be seen as choices over present and future, once current 7 A is thus a set of product measures. But there is no loss of generality: the Pareto Axiom implies that the social planner is indifferent toward correlated individual outcomes. Only marginal distributions on individual components matter, and not the full distribution. 16

18 uncertainty is resolved. They are homeomorphic to choices on lotteries in which first period consumption is non-stochastic: each element of this subset is the product measure of a one-point measure on x X, δ x, and of a measure a A (describing the uncertain future). It writes δ x a, it is an element of A and naturally represents an element of X A. The prominent approach to choices in stochastic environments involves assuming that the decision maker is an expected utility maximizer. This is guaranteed by von Neumann and Morgenstern (vnm) axioms of choice under uncertainty, thus we shall refer to vnm preferences. If the decision maker is vnm, there exists a continuous Bernoulli utility function U defined on X up to a positive affine transformation such that, for all a and â in A, a â E a U EâU where E a U = X U da is the expectation with respect to measure a. We denote U i y i t,t a Bernoulli function for individual i at time t given a payoff history yi t Y t. U i y t) = E t i,t(ai a i t U i represents individual preferences on A. Similar notation are yt i,t used to describe the planner s preferences on A. In this setting, Proposition 1 has the following implication: Proposition 3 Suppose that the social aggregation is Paretian. Assume that the planner s preferences satisfy vnm axioms of choice under uncertainty. Then social preferences are consistent and history independent if and only if individuals are expected utility maximizers with additively separable Bernoulli utility functions and the social Bernoulli function is a sum of individual ones. Proof. See the appendix. Proposition 3 bears some similarities to Harsanyi s (1955) theorem which states that a Paretian aggregation of individual vnm utilities can only be repre- 17

19 sented by a sum of individual Bernoulli utility functions. The additional result is that, when the planner s objective is history independent, the Bernoulli utility functions of all agents must be additively separable. This condition of additive separability has implications similar to those found in the deterministic case. Indeed, individuals and the planner s preferences should then not only be independent of the past but also risk independent of the future. But the additive separability of Bernoulli utility functions also has consequences in terms of correlation aversion. 8 Indeed, it implies that the decision makers are indifferent to the correlation of their intertemporal outcomes. On the contrary, many papers have underlined that correlation aversion seems to have realistic empirical implications (see Epstein and Tanny, 1980, or Eeckhoudt, Rey and Schlesinger, 2007). Lastly, note that the extension of Proposition 2 to the vnm framework would imply that the Bernoulli utility functions of all individuals and of the planner should be of the forms described in Equations (4) and (6). 5.3 Stochastic dynamic choice: K-P preferences Kreps and Porteus (1978) have provided an axiomatic to take into account the timing of the resolution of uncertainty in stochastic dynamic problems. It was extended to the infinite dimension case by Epstein and Zin (1989). This involves defining temporal lotteries that describe when the uncertainty is resolved. vnm preferences corresponds to the special case in which timing does not matter. The construction of the set of temporal lotteries in an infinite time-horizon framework is a complicated exercise. But it has been done by Epstein and Zin (1989), when the set of current payoff X is a compact subset of the positive real line, R +. Let A be the set of temporal lotteries. They show that it is a separable metric space homeomorphic to X M(A), where M(A) is the set of 8 The concept was brought in the economic literature by Richard (1975) and further discussed in Epstein and Tanny (1980). 18

20 Borel probability measures on A. An interesting subset of A is the one whose elements are of the form x δ a : they select a particular temporal lottery for the next period. Clearly, X A is homeomorphic to this set. Each period, decision makers choose between temporal lotteries. But under Kreps and Porteus axioms, history independent preferences on this set have a very specific form: 9 Definition 4 A decision maker has history independent K-P preferences on temporal lotteries if there exist functions u t : X A R and U t : A R defined recursively for each t N, y Y t, x t X and a t+1 A by: U t (x t, a t+1 ) = u t (x t, E at+1 U t+1 ) (7) and such that: a t t â t E at U t Eât U t (8) As before, E a represents the expectation taken with respect to measure a. The form of the aggregator function defining preferences on X A in Equation (7) clearly implies that K-P preferences are consistent. Kreps and Porteus (1978) prove that if the aggregator function u t (.) in Equation (7) is linear, the decision maker is indifferent to the timing of the resolution of uncertainty. If it is so, we are back to the vnm construction: the decision maker is an expected utility maximizer, whose preferences over lotteries are defined by Equation (8). Proposition 1 has the following implication: 9 We call K-P preferences preferences satisfying axioms defined in Kreps and Porteus (1978). They did not assume history independence in their work. But the history independent specification is the one most preferred in the literature, and it is in particular the one adopted by Epstein and Zin (1989). 19

21 Proposition 4 Assume that the social planner has K-P preferences. A Paretian aggregation of individual preferences over temporal lotteries can be history independent if and only if all individuals and the planner have vnm preferences, fulfilling the conditions given in Proposition 3. Proof. See the appendix. All individuals and the planner should be indifferent to the timing of the resolution of uncertainty if we want an history independent social objective. They must also be indifferent to correlated intertemporal outcomes due to the conditions in Proposition 3. This result excludes the C.E.S. aggregator function suggested in Epstein and Zin (1989). Although this work and others have provided many illuminating insights on individual intertemporal choices, the form of preferences it suggests cannot result from a Paretian aggregation, unless it corresponds to the standard additively separable vnm case. Proposition 4 has important consequences for social choices. As explained by Gottshalk and Spolaore (2002), Kreps and Porteus preferences can be used to represent two aspects of economic mobility: reversal and origin independence. Reversal means that society prefers an equalization of multi-period income and therefore supports equalizing reversals of social positions. Origin independence means that the society prefers situations where the prospects of individuals are not correlated to their past experience. The additive structure imposed by Proposition 4 rules out these two sources of social preference for mobility. Since individuals are indifferent to the timing of the uncertainty and since the social objective is a sum of additively separable vnm utilities, there is no preference for origin independence: correlations with past outcomes and the timing of the revelation of social positions do not matter. Additive vnm utilities also mean no preference for long term equalization à la Atkinson and Bourguignon (see Section 5.1). Once again, this feature of social preferences is likely to be deemed ethically 20

22 unappealing. 6 Conclusion We have found stringent conditions for a Paretian aggregation of intertemporal utilities to be history independent. They bear both on individual preferences and on the permitted aggregations. Individual utilities must be additively separable. We know that this has strong implications in terms of time-independence, risk-independence etc. In the stochastic context, it also means indifference to the intertemporal correlation of outcomes. When considering temporal lotteries, this results in indifference to the timing of the resolution of uncertainty. If we want a stationary and Paretian social objective, the only class of admissible individual utilities are those exhibiting exponential discounting. In addition, people must have the same rate of time discounting, canceling out an important dimension of the heterogeneity in preferences. The form of social aggregation is also constrained. The social objective must be a Utilitarian aggregation of the additive individual utilities. As a consequence, the social welfare function also has to be additive, with consequences similar to those stated for individual preferences. More importantly, the society treats each period of an individual s life separately, taking no account of the fact that some people may be systematically disadvantaged. This results in a form of indifference to inequality and mobility. Our results can be read in either of two ways. We might first be left with the impression that the restrictions uncovered in the paper should be accepted, despite their lack of realism. Indeed, utilitarian aggregations of additively separable utilities are commonplace in the economic theory. However, the applications to specific setting have highlighted that such a construction rules out significant phenomena such as endogenous discounting, correlation aversion, an so forth. Many 21

23 welfare functions considered in the macroeconomic literature, in particular, recursive preferences as studied in Beals and Koopmans (1969), Uzawa (1968) or Epstein and Zin (1989) would be excluded, for they cannot embody the decisions made by a history independent paretian social planner, except for their additive special case. Last, the additive criteria have ethically disputable consequences in terms of preference for mobility and intertemporal inequality. This suggests that a second path should be found, allowing to preserve some of those features. This second path entails abandoning the assumption of history independence. The property has indeed no particular normative appeal. But then the dependence must be carefully modelled. For instance, we can try to aggregate the recursive preferences of heterogeneous individuals. Lucas and Stockey (1985) have shown that Paretian equilibria can be computed in that case. Their results also suggest that, when considering a Utilitarian aggregation, the dependence of the past takes the form of changing weights in the social welfare function. Some research could be undertaken in that promising direction. Hopefully, we could obtain social objectives indicating how past inequalities should modify current choices. Appendix Proof of Proposition 1 A Utilitarian aggregation of additive preferences is clearly consistent and history independent. To prove the necessity part of Proposition 1, we use results by Debreu (1959) and Gorman (1968). But we need first to introduce some definitions. Consider a continuous preference ordering on a product space S = S 1 S 2 S M. The set of factors indexes is J = {1,, M}. For a subset K of J, S K = k K S k. For x S, we denote x j the component in S j. 22

24 Definition: The factors S K are said to be separable for the preference ordering if, for any x, x, y and y in S such that x k = x k and y k = y k for all k K and x j = y j and x j = y j for all j J \ K: x y x y We also need a minimum sensitivity requirement called essentiality: Definition: The factor S j is essential for preference ordering if there exist x and y such that x j y j and x k = y k for all k J \ {j} and: x y Two important results can now be stated. Proposition (Debreu, 1959): Assume that for any subset K of J the factors S K are separable for the preference ordering. If there exist at least three factors which are essential, then there exist continuous functions u j (.) on each S j such that, for any x and y in S: x y U(x) = j J u j (x j ) j J u j (y j ) = U(y) To obtain separable sets, a second result can be used: Proposition (Gorman, 1968): Let (I, K, L, M) be a partition of the set of indexes J. If S I K and S L K are separable and essential for the preference ordering, then S I, S K, S L, S I L and S I K L are separable and essential for the preference ordering. Consider social preferences S y t, t on X A. By history independence it can be expressed S t and we know that the factors A are separable for S t. The Pareto 23

25 axiom combined with the assumption of self-regarding preferences implies that, for any i I, y Y t, if a and â in A such that a i â i and a j = â j, for j I \{i}, U S y,t(a) U S y,t(â) U i y i,t (ai ) U i y i,t (âi ). Thus X i A i is separable for S t. If we impose the mild minimum sensitivity requirement that each X i A i and A are essential, 10 by Gorman s (1968) result, all of the following are separable and essential: X i, A i, X i A j, X i A i A j. By appropriate unions and intersections of such sets, we obtain that any subset of factors in X A is separable and essential for S t. By Debreu s (1959) theorem, there must exist real-valued functions u i t(.) on X and W i t (.) on A such that, for any (x t, a t+1 ) and (ˆx t, â t+1 ) in X A: (x t, a t+1 ) S t (ˆx t, â t+1 ) i I u i t(x i t) + i I W i t (a i t+1) i I u i t(ˆx i t) + i I W i t (â i t+1) By the Pareto Axiom, U i t (x i t, a i t+1) = u i t(x i t)+w i t (a i t+1) must represent individual i s preferences on X A at time t. Moreover U S t (x t, a t+1 ) = i I U i t (x i t, a i t+1): the Utilitarian aggregation of preferences represents social preferences. Individual preferences are represented by U i t (x i t, a i t+1) : they must be history independent. They must also be consistent in view of the following lemma: Lemma 1 If social preferences are Paretian and consistent, all individuals must have consistent preferences. Proof. Take a and â in A such that a i â i and a j = â j, for j I\{i}. By the Pareto Axiom and the consistency of social preferences, we obtain that, for any i I, y i Y i t, x i X, a i and â i in A: U i y i,t (xi, a i ) U i y i,t (xi, â i ) U i (y i,x i ),t+1 (ai ) U i (y i,x i ),t+1 (âi ) This is the definition of individual preferences being consistent. Thanks to Lemma 1, we know that W i t (a i t+1) must represent individual i s pref- 10 By Pareto s principle, this would be the case if individuals are not indifferent between all options and if the future matters. 24

26 erences on A at time t+1. In principle, we could have W i t (a i t+1) = φ i t[ U i t+1 (a i t+1) ], with φ i t(.) a strictly increasing continuous function. But, for social preferences to be consistent, we need W i t (a i t+1) = δ t U i t+1(a i t+1), with δ t R +. Thus, individual preferences are additive. Consequently, the planner also has additive preferences on X A. Proof of Proposition 2 The sufficiency part of the proposition is obvious. For the necessary part, let us first prove the next lemma: Lemma 2 If social preferences are Paretian and stationary, all individuals must have stationary preferences. Proof. Take a and â in A such that a i â i and a j = â j, for j I\{i}. Using the Pareto Axiom and the stationarity of social preferences we obtain, for any i I, x i X, a i and â i in A: U i 1(a i ) U i 1(â i ) U i 1(x i, a i ) U i 1(x i, â i ) Individual preferences are stationary. Social preferences are stationary, hence history independent and consistent. Thus Proposition 1 applies and individual preferences are additive. The only additive and stationary preferences are those exhibiting exponential discounting: there exist functions u i and U i and a constant δ i R + such that, for any i I, x i X and a i A, U i (x i, a i ) = u i (x i ) + δ i U i (a i ) represents individual i s preferences. U S (x, a) = i I U i (x i, a i ) represents social preferences on X A. But for this aggregation to be stationary, we necessarily need δ i = δ, for all i I. Proof of Proposition 3 Let us begin by proving the following lemma: 25

27 Lemma 3 If social preferences are Paretian and vnm, all individuals must be expected utility maximizers. Proof. Take a and â in A such that a i â i, and a j = â j = δ ( x, x, ), with x X, for j I \ {i}. Pareto s axiom implies that, for any i I, y Y t, a i and â i in A, a S y,t â a i i y i,t âi. But the planner has vnm preferences so that a S y,t â E a iu S y,t EâiU S y,t. Thus, a i i y i,t âi E a iu S y,t EâiU S y,t. Consequently, i is an expected utility maximizer with a Bernoulli utility function U y i,t(x i t, x i t+1, ) = ) U y,t (( x, x, ),, ( x, x, ), (x i t, x i t+1, ), ( x, x, ),, ( x, x, ). Under the assumptions of Proposition 3, we know by Proposition 1 that there exist real-valued functions u i t(.) on X and W i t (.) on A such that i I ui t(x i t) + i I W i t (a i t+1) represents social preferences on X A at time t. We also know that Wt i (a i t+1) represents individual i s preferences on A at time t + 1, and so is ( ) ordinally equivalent to E a i t+1 U i t+1. Wt i (a i t+1) = φ i t E a i t+1 U i t+1, where φ i t(.) is a continuous increasing function. By consistency of social choices, i I W t i (a i t+1) = ( ) E a i t+1 U i t+1 must be ordinally equivalent to E at+1 U S t+1 : E at+1 U S t+1 = i I φi t ψ ( i I φi t(e at+1 U i t+1) ) with ψ(.) a continuous increasing function. By linearity of expectation, this is possible only if functions ψ and φ i t(.) are affine functions. Then W i t (a i t+1) = E a i t+1 U i t+1 with U i t+1 a Bernoulli utility function for the individual (not necessarily the one used before). U S t+1(.) is a sum of those. t+1χ i = (x i t+1,, x i t+t, ) is a program for the infinite future, belonging to X. It only remains to prove that u i t(x i t) + U i t+1( t+1 χ i ) is a Bernoulli utility function used to represent i s preferences at time t. To do so, consider preferences for (x i t, t+1χ i ) X, when there is no uncertainty. They are represented by u i t(x i t) + U i t+1( t+1 χ i ) and by U i t(x i t, t+1 χ i ). Thus u i t(x i t) + U i t+1( t+1 χ i ) = ( ) U i t(x i t, t+1 χ i ), with ϕ i t(.) a continuous increasing function. i I Ui t(x i t, t+1 χ i ) ϕ i t is a Bernoulli function for the planner representing his preferences on i I X. 26

28 i I ui t(x i t) + i I Ui t+1( t+1 χ i ) = ( ) i I ϕi t U i t(x i t, t+1χ i ) preferences. We thus obtain the functional equation: i I ϕ i t ( ) ( ) U i t(x i t, t+1 χ i ) = Ψ U i t(x i t, t+1 χ i ) i I also represents those whose solution is that the functions ϕ i t(.) and Ψ(.) are affine. Thus u i t(x i t) + U i t+1(c i t+1), an affine transform of a Bernoulli utility function, is a Bernoulli utility function for individual i. Proceeding by induction, we end up with U i t( t+1 χ i ) = τ=t ui τ(x i τ) : the individual Bernoulli utility functions are additively separable. Proof of Proposition 4 First, we have Lemma 4, similar to Lemma 3: Lemma 4 If social preferences are K-P and the planner is Paretian, all individuals must have K-P preferences. Proof. Similarly to the proof of Lemma 3, consider two temporal lotteries that differ for one individual only. The use of the Pareto axiom shows that the individual must have K-P preferences. Individual preferences are thus defined on X A by equation (7). But Proposition 1 implies that for each individual the aggregator function in this equation is linear. We know that in that case individuals are expected utility maximizers. The same reasoning is true for social preferences. Then Proposition 3 applies. References Atkinson, A.B., Bourguignon, F. (1982). The comparison of multi-dimensioned distributions of economic status, Review of Economic Studies, 49,

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