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Social Choice and Welfare

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01-12-2005 | Original Paper

The news of the death of welfare economics is greatly exaggerated

Authors: Marc Fleurbaey, Philippe Mongin

Published in: Social Choice and Welfare | Issue 2-3/2005

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Abstract

The paper reexamines the controversy about Bergson–Samuelson social welfare functions (BSFs) that took place between welfare economists and social choice theorists as a consequence of Arrow’s (1951) impossibility theorem. The 1970’s witnessed a new version of the theorem that was meant to establish that BSFs “make interpersonal comparisons of utility or are dictatorial.” Against this, Samuelson reasserted the existence of well-behaved “ordinalist” BSFs and generally denied the relevance of Arrovian impossibilities to welfare economics. The paper formalizes and reassesses each camp’s arguments. While being also critical of Samuelson’s, it eventually endorses his conclusion that welfare economics was left untouched by the controversy. It draws some connections of BSFs with contemporary normative economics.

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On the Compensation Principle, see Chipman and Moore (1978). More recent investigations include Gravel (2001) and Suzumura (1999).

This statement is borrowed from Pollak (1979, p. 76). Parks (1976) has a different formulation in terms of utility values.

As a matter of attribution, it appears that Kemp and Ng on the one hand, Parks on the other worked independently and simultaneously, while Pollak was stimulated by the former’s work and did not initially know of the latter’s. The argument in Kemp and Ng involved an irrelevant topological framework, and had many loose ends. Pollak extracted the neat Theorem 2 from this jumble. Parks had already reached Theorem 2, but his paper is terse, and does not draw the connections with Theorem 1 in the same useful way as Pollak does.

Arrow (1963, ch. VI) initiated the study of economic domains in social choice theory, but full proofs had to await the 1980’s. For a survey, see Le Breton and Weymark (1996).

Turning May’s characterization into an Arrovian theorem, Sen (1970, p. 73) proves that no F can satisfy the universal domain condition, neutrality, positive responsiveness (an interprofile condition which is related to (WP)), and anonymity (which is a strong denial of (D)). This may be the first impossibility theorem involving neutrality in its premisses.

Blackorby et al. (1984) diagrammatic analysis of social choice uses a similar argument. See also Blackorby et al. (1990).

It is not always the case that the maximal domain should be selected when there is no reason to select any particular domain. For a discussion in the context of choice functions, see Sen (1971, 1982, p. 48–49) and Mongin (2000, p. 92–93).

Sadly, Little’s (1999, p. 17–18) late retraction does not show that he has taken advantage of the intervening clarifications.

See also Samuelson’s rebuttal of Lange and Fisher: “Lange, Fisher, and others have contended that measurable utility, while superfluous from the standpoint of positivistic behavioristic description, is necessary for the purpose of a normative science of welfare economics... It is well to point out that this is not at all necessary” (1947, p. 173).

A function $U_{i} \in \mathbb{R}^{X}$ represents R _i on X if, for all x, y ∈ X, xR _i y iff U _i(x)≥U _i(y). The vector-valued function $U = {\left( {U_{i} } \right)}_{{i \in N}}$ is said to represent ${\left( {R_{i} } \right)}_{{i \in N}}$ when U _i represents R _i for each i.

This formula can be found in Arrow’s comment of Samuelson (1983, p. 22).

Compare this statement with Pazner and Schmeidler (1978, p. 679).

We read no further advance in Bergson’s (1976) latest retrospective paper.

This text was regarded by Samuelson as an authoritative restatement of the new welfare economics; see his preface.

The proof of Theorems 1_f and 2_f from Theorems 1 and 2 uses the fact that the latter remain true when ${\user1{\mathcal{D}}}^{F}$ is restricted to the set of those orderings on X which can be represented by a utility function. Fishburn (1970, p. 27–28) states that there exists U _i representing R _i if and only if P _i is order-dense on the quotient of X by the indifference relation I _i.

Interestingly, Hammond (1976) derives (UN)_f from (UI)_f and a weaker form of neutrality. He proves a variant of Theorem 4 stated below.

We choose this particular condition because it is simple and can be satisfied on economic domains. Bordes et al. (1996) investigate more basic conditions, and so does Weymark (1998).

It is well-known that on relevant economic domains, (PI)_f follows from (WP)_f and a continuity condition (see, e.g., Suzumura, 2001).

Compare this technical analysis with Roberts’ (1980) and d’Aspremont’s (1985). In ways slightly different from ours, they tighten the links between Arrow’s multi-profile theorem and his followers’ single-profile version.

Blackorby et al. (1990, p. 283) reach a similar conclusion.

For clarity of Fig. 4, the two pairs of allocations (x, y) and (z, w) use different amounts of resources, but it is easy to adapt the example to a case without free disposal.

Similarly, consider again the economic example of the distribution of Ω among n individuals. Suppose alternatives are redescribed in this way:

$$\widehat{x} = {\left( {x_{1} , \ldots ,x_{n} ,\Omega ,U_{1} * {\left( {x_{1} } \right)}, \ldots ,U_{n} * {\left( {x_{n} } \right)}} \right)}.$$

If one believes that E* is the proper ethical criterion for this example, then one must admit that $\widehat{x}$ contains all the relevant information. But it still holds that individual i weakly prefers $\widehat{x}$ to $\widehat{y}$ if and only if x _i R _i y _i. Therefore (I) applied to individual preferences over $\widehat{x}$, $\widehat{y}$ is just the same as (I) applied to the initial x, y, and is just as restrictive and questionable.

See, e.g.:“I shall argue that the Arrow result is much more of a contribution to the infant discipline of mathematical politics than to the traditional mathematical theory of welfare economics” (Samuelson 1967, p. 168).

Samuelson is justified in attributing to Bergson a rejection of independence. However, Bergson’s (1954, p. 244–245; see also 1966, p. 75–76) criticism relied on the concept of need, and virtually ignored the point that Samuelson is making here, i.e., that individual preferences about third allocations are relevant in the evaluation of two allocations.

Pollak’s conclusion coincides with Samuelson’s, but the main argument he provides is oddly the ineffective Bergsonian one “that the Bergson–Samuelson social welfare function reflects the judgments of an ethical observer” (1979, p. 86).

In a stronger version called “independence of irrelevant preferences”, Mayston requires only the intersection of B _i(x, y) with a reference path Z crossing all indifference curves to be the same in ${\overrightarrow{R}}$ and ${\overrightarrow{{R^{\prime } }}} $.

Similarly, Fleurbaey and Maniquet (1996) were not aware of either Hansson, Mayston or Pazner when they proposed in their turn to weaken the independence condition.

See Mongin (1999, 2002).

Witness this remarkable statement: “I now feel... that the austerity imposed by this condition is stricter than desirable; in many situations we do have information on preferences for nonfeasible alternatives. It can certainly be argued that when available this information should be used in social choice... The potential usefulness of irrelevant alternatives is that they may permit empirically meaningful interpersonal comparisons” (Arrow 1967, p. 19, our emphasis).

Studies of tax issues in Fleurbaey and Maniquet (2002a,b) and Fleurbaey (2005) exemplify both this kind of heuristics and some of its preliminary results.

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Title: The news of the death of welfare economics is greatly exaggerated
Authors: Marc Fleurbaey
Philippe Mongin
Publication date: 01-12-2005
Publisher: Springer-Verlag
Published in: Social Choice and Welfare / Issue 2-3/2005
Print ISSN: 0176-1714
Electronic ISSN: 1432-217X
DOI: https://doi.org/10.1007/s00355-005-0010-1

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Other articles of this Issue 2-3/2005

The launching of ‘social choice and welfare’ and the creation of the ‘society for social choice and welfare’

The intellectual contribution of Condorcet to the founding of the US Republic 1785–1800

Distribution and politics: a brief history and prospect

An interview with I. M.D. Little

The pre-history of Kenneth Arrow's social choice and individual values

Condorcet's Social Mathematic, A Few Tables

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