Infinite Utility Streams Research Articles

Ordering infinite utility streams comes at the cost of a non-Ramsey set

The existence of a Paretian and finitely anonymous ordering in the set of infinite utility streams implies the existence of a non-Ramsey set (a non-constructive object whose existence requires The Axiom of Choice). Therefore, each Paretian and finitely anonymous quasi-ordering either is incomplete or does not have an explicit description. Hence, the possibility results of Svensson [Svensson, L.G., 1980. Equity among generations. Econometrica 48, 1251–1256] and of Bossert et al. [Bossert, W., Sprumont, Y., Suzumura, K., 2007. Ordering infinite utility streams. Journal of Economic Theory 135 (1), 579–589] do require The Axiom of Choice.

Ordering Infinite Utility Streams Comes at the Cost of a Non-Ramsey Set

The existence of a Paretian and finitely anonymous ordering in the set of infinite utility streams implies the existence of a non-Ramsey set (a nonconstructive object whose existence requires the axiom of choice). Therefore, each Paretian and finitely anonymous quasi-ordering either is incomplete or does not have an explicit description. Hence, the possibility results of Svensson (1980) and of Bossert, Sprumont, and Suzumura (2006) do require the axiom of choice.

$${\mathcal{Q}}$$ -anonymous social welfare relations on infinite utility streams

This paper studies a class of social welfare relations (SWRs) on the set of infinite utility streams. In particular, we examine the SWRs satisfying Q-Anonymity, an impartiality axiom stronger than Finite Anonymity, as well as Strong Pareto and a certain equity axiom. First, we characterize the extension of the generalized Lorenz SWR by combining Q-Anonymity with Strong Pareto and Pigou-Dalton Equity. Second, we replace Pigou-Dalton Equity with Hammond Equity for characterizing the extended leximin SWR. Third, we give an alternative characterization of the extended utilitarian SWR by substituting Incremental Equity for PigouDalton Equity.

On the impossibility of representing infinite utility streams

We introduce a new Pareto-type criterion for social welfare functions over infinite utility streams that is not necessarily sensitive to increments in just a finite number of components. We show that there is no social welfare function that satisfies both this criterion and Diamond’s equity condition simultaneously. With our result, we extend the impossibility theorem of Basu and Mitra. Moreover, we show that, even under a weaker version of equity related to Zame’s intergenerational equity condition, the impossibility results are obtained as well.

On the continuity of ethical social welfare orders on infinite utility streams

In this paper we study the extent to which ethical social welfare orders on infinite utility streams can be continuous. For a class of metrics, we show that ethical preferences can be continuous if and only if the continuity requirement is in terms of a metric which satisfies a simplex condition. This condition requires that the distance from the origin to the unit simplex in the space of utility streams be positive. We use this characterization result to establish that the metric used by Svensson (Econometrica 48:1251–1256, 1980) induces the smallest topology for which there exist continuous ethical preferences.

Continuity and egalitarianism in the evaluation of infinite utility streams

There exists a utilitarian tradition a la Sidgwick of treating equal generations equally. Diamond showed that there exists no social evaluation ordering over infinite utility streams in the presence of the Pareto principle, the Sidgwick principle, and continuity. Instead of requiring the Sidgwick principle of procedural fairness, we focus on two principles of distributional egalitarianism along the line of the Pigou–Dalton transfer principle and the Lorenz domination principle, and show that there exists no social evaluation relation satisfying one of these egalitarian principles and the weakened continuity and rationality axioms even in the absence of the Pareto principle.

Patience in some non-additive models

Our aim is to give an axiomatization of preferences over infinite consumption streams. At first we adopt the additive case, and give a characterization of preferences which satisfy patience [Marinacci, M., 1998. An axiomatic approach to complete patience and time invariance. Journal of Economic Theory 83, 105–144] or equivalently what Diamond [Diamond, P.A., 1965. The evaluation of infinite utility streams. Econometrica 33, 170–177] named equal treatment of all generations and then, focus on stationary additive preferences. It appears that this class of functionals contains the discounting functionals axiomatized in Koopmans [Koopmans, T.C., 1972. In: McGuire, C.B., Radner, R. (Eds.), Representations of Preference Orderings Over Time. Decision and Organization, North-Holland, Amsterdam] and what is known as Banach-Mazur limit functionals. These results are extended to non-additives preferences where similar results are generalized and naive patience receives a positive treatement through the liminf criterion.

On the Possibility of Continuous, Paretian and Egalitarian Evaluation of Infinite Utility Streams

There exists a utilitarian tradition a la Sidgwick of treating equal generations equally in the form of anonymity. Diamond showed that no social evaluation ordering over infinite utility streams satisfying the Pareto principle, Sidgwick's equity principle, and the axiom of continuity exists. We introduce two versions of egalitarianism in the spirit of the Pigou-Dalton transfer principle and the Lorenz domination principle, and examine their compatibility with the weak Pareto principle in the presence of a semi-continuity axiom. The social evaluation relation is not assumed to be either complete or transitive, yet Diamond's impossibility strenuously resurfaces.

On the equity-efficiency trade off in aggregating infinite utility streams

We compare two different approaches to aggregating infinite utility streams, following some principles of equity and efficiency. We find that the consequentialist equity axiom, Hammond Equity for the Future, is more demanding than its procedural equity counterpart, Anonymity.

On the impatience implications of Paretian social welfare functions

In this paper, we investigate the impatience implications resulting from the assumption of existence of a Paretian social welfare function (SWF) aggregating infinite utility streams. We show, for very general program spaces, that the set of utility streams, at which the SWF exhibits impatience, has the power of the continuum. In the context of a more special program space, which has figured prominently in the literature, we establish that this set is dense, so that even if there is a point in the program space at which the SWF does not exhibit impatience, there are points close to it at which it does. If the Paretian SWF is continuous (in the sup metric), we show that impatience is generic: the collection of points, at which the SWF does not exhibit impatience, is a closed, nowhere dense set.

On the Extension of the Utilitarian and Suppes–Sen Social Welfare Relations to Infinite Utility Streams

Extensions of a utilitarian and a Suppes–Sen grading principle defined on infinite utility streams are characterized with a stronger notion of Anonymity and without any consistency postulate. The relative merits of the Extended Utilitarian relation are discussed and its rankings are compared with those of the overtaking criterion and the Basu–Mitra Utilitarian relation.

Ordering infinite utility streams

We reconsider the problem of ordering infinite utility streams. As has been established in earlier contributions, if no representability condition is imposed, there exist strongly Paretian and finitely anonymous orderings of intertemporal utility streams. We examine the possibility of adding suitably formulated versions of classical equity conditions. First, we provide a characterization of all ordering extensions of the generalized Lorenz criterion as the only strongly Paretian and finitely anonymous rankings satisfying the strict transfer principle. Second, we offer a characterization of an infinite-horizon extension of leximin obtained by adding an equity-preference axiom to strong Pareto and finite anonymity.

Utilitarianism for infinite utility streams: A new welfare criterion and its axiomatic characterization

A definition of a utilitarian social welfare relation (SWR) for infinite utility streams is proposed. Such a relation is characterized in terms of the Pareto, Anonymity and Partial Unit Comparability Axioms. The merits of the utilitarian SWR, relative to the more restrictive SWR induced by the overtaking criterion, are examined.

Formal Welfarism and Intergenerational Equity

Intergenerational justice is a matter that should primarily concern the present generation, since the individuals living now are those to take immediate decisions affecting generations that will be living in the future, and even in the far future, as we know, for example, from the exhaustibility of some resources or from the long-term effects of pollution such as global warming. Of course, each future generation will become ‘present’ at some point in time, and the reasoning followed for the present ‘present generation’ about intergenerational justice could be repeated at that point in time. But, to develop this reasoning, each present generation should have a representation of future generations’ interests. In that respect, a simple formulation of the problem that has been extensively analyzed consists in trying to find, under equity and efficiency conditions, an ordering of the set of possible ‘infinite utility streams’, that is, of the set of possible infinite sequences of utility levels attached to the successive generations starting with the present generation. In such a formulation, the welfare of each generation is represented by a single utility level, as if a generation were composed of a single individual or of a cohort of identical individuals with identical allocation.

Aggregating Infinite Utility Streams with InterGenerational Equity: The Impossibility of Being Paretian

It has been known that, in aggregating infinite utility streams, there does not exist any social welfare function, which satisfies the axioms of Pareto, intergenerational equity, and continuity. We show that the impossibility result persists even without imposing the continuity axiom, and in frameworks allowing for more general domains of utilities than those used in the existing literature.

Aggregating Infinite Utility Streams with Inter-generational Equity

It has been known that, in aggregating infinite utility streams, there does not exist any social welfare function, which satisfies the axioms of Pareto, inter-generational equity and continuity. We show that the impossibility result persists even without imposing the continuity axiom. Hence, the problem of accommodating inter-generational equity is more obstinate than previously supposed. The paper goes on to explore the scope for obtaining possibility results by weakening the Pareto axiom and placing restrictions on the domain of utilities.

Intertemporal objective functions: Strong pareto versus anonymity

This paper continues work begun by Koopmans: how to order infinite utility, consumption, or payoff streams; and, in particular, how to provide an axiomatic basis for ordering them. Diamond's anonymity axiom is compatible with dictatorship and is therefore too weak. We motivate a form of anonymity that gives an equal weight to equally large coalitions, and investigate its implications. Medial limits are shown to satisfy this version of anonymity. A joint characterization reduces the choice between discounting rules and medial limits to the choice between anonymity and the strong Pareto principle. By means of an example we show that discounting optima may differ from anonymous optima in an essential way.

Continuity and equity with infinite horizons

In an infinite dimensional space, e.g. the set of infinite utility streams, there is no natural topology and the content of continuity is manipulable. Different desirable properties induce different topologies. We consider three properties: effectiveness. l1-summability and equity. In view of effectivity, the product topology is the most favourable one. The strict topology is the largest topology for which all the continuous linear maps are l1-summable. However, both topologies are myopic and conflict with the principle of equity. In case equity is desirable, the sup topology comes forward.

Time-neutrality and linearity

This paper provides a definition of a time-neutral evaluation on the set of bounded infinite utility streams and characterizes time-neutral linear evaluations.

The Evaluation of Infinite Utility Streams

The Evaluation of Infinite Utility Streams