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The Axiomatic Approach to the Ranking of Infinite Streams

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The Economics of the Global Environment

Part of the book series: Studies in Economic Theory ((ECON.THEORY,volume 29))

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Abstract

The history of the axiomatic approach to the ranking of infinite streams starts with Koopmans (1960) characterization of the discounted utilitarian rule. This rule, however, meets Chichilnisky’s axiom of dictatorship of the present and puts future generations offside. Recently, Lauwers (2010a) and Zame (2007) have uncovered the impossibility to combine in a constructible way the requirements of equal treatment, sensitivity, and completeness. This contribution presents and discusses different axioms proposed to guide the ranking of infinite streams and the criteria they imply. The literature covered in this overview definitely points towards a set of meaningful alternatives to discounted utilitarianism.

I thank Graciela Chichilnisky and Armon Rezai for inviting me to write this chapter. I am also grateful for the comments from a referee, Tom Potoms, and Luc Van Liedekerke.

24 juli 2014.

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Notes

  1. 1.

    In his review on intergenerational equity, Asheim (2010, Sect. 3.2) coins this result as the Lauwers-Zame impossibility theorem.

  2. 2.

    I want to mention already here that the combination of continuity with respect to the sup-topology and representability does not guarantee that the ranking rule is constructible. The other way around, the representation of a non-constructible ordering, is in itself a non-constructible object.

  3. 3.

    Ferejohn and Page (1978) have shown that unrestricted domain, Pareto, Arrovian independence, and stationarity results in dictatorship of the first generation. This result has been strengthened by Packell (1980).

  4. 4.

    Section 3 provides a simple example to indicate that paths optimal with respect to a sustainable-equitable approach might differ substantially from optima generated by the discounted utilitarian rule.

  5. 5.

    See also Koopmans (1960, 1965, 1972a) and Koopmans et al. (1964).

  6. 6.

    For simplicity, \(Y=\mathbb {R}\), \(x_t\) is the consumption of generation t, and the set \(\ell _\infty \) of bounded streams takes the role of X.

  7. 7.

    Koopmans considers infinite streams of vectors instead of scalars. The axiom of monotonicity is a one-dimensional version of Koopmans’ axiom.

  8. 8.

    Diamond (1965) follows Debreu (1954) to prove this result.

  9. 9.

    This motivation, however, is wrong in case the continuous social welfare function represents a non-constructible order.

  10. 10.

    See also Fleurbaey and Michel (2003, Sect. 3.4).

  11. 11.

    Different axiomatizations are obtained by Lauwers (1997c), Bleichrodt et al. (2008), and Asheim et al. (2012).

  12. 12.

    The factor \((1-\alpha )\) in the definition of D ensures that \(D(x,x,\ldots ,x,\ldots )=u(x)\). Hence, the weights with which the \(u(x_t)\) are multiplied add up to 1.

  13. 13.

    The map \(x\mapsto (1-\beta )(x_1+\beta \, x_2+\cdots +\beta ^{t-1}x_t+\cdots )\) obtains a maximal value, equal to 0.3025, in one of the streams of type \(t^n\); while the stream \(u^\infty \) obtains a lower value of 0.25.

  14. 14.

    Dubey (2011) and Dubey and Mitra (2011) investigate the role of the set Y of possible utility levels and refine the results of Lauwers (2010a) and Zame (2007).

  15. 15.

    We refer to Suppes (1966), Sen (1971), Hammond (1976), d’Aspremont and Gevers (1977), and Shorrocks (1983). Bossert and Weymark (2004) provide an excellent overview.

  16. 16.

    Note the similarity with the decisive sets in Arrow’s impossibility theorem. See also Fleurbaey and Michel (2003).

  17. 17.

    Basu and Mitra (2007) discuss this example.

  18. 18.

    Demichelis et al. (2010) study axioms of anonymity in combination with strong Pareto and stationarity.

  19. 19.

    A non-Ramsey set is a subset \(\mathcal N\) of the collection \(\mathbb {N}_\infty \) of all infinite subsets of \(\mathbb {N}\) such that for each element J in \(\mathcal N\) the collection of infinite subsets of J intersects both \(\mathcal{N}\) and its complement \(\mathbb {N}_\infty -\mathcal{N}\). The technique developed in Lauwers (2010a) to define non-Ramsey sets has been used by Dubey and Mitra (2013) to show that a complete ranking that combines strong Pareto and Hammond equity (or the strict transfer principle) is non-constructible. See also Dubey (2011), Dubey and Mitra (2011, 2012), and Banerjee and Dubey (2013).

  20. 20.

    Doyen and Martinet (2012) apply the maximin rule in a general dynamic economic model.

  21. 21.

    Asheim and Zuber (2013) study the behavior of the rank-discounted utilitarian rule as \(\beta \) goes to zero and show the convergence of R towards a strongly anonymous leximin relation.

  22. 22.

    Also Banerjee (2006a), Asheim et al. (2007), Alcantud and García-Sanz (2010), Dubey and Mitra (2013) consider Hammond equity for the future.

  23. 23.

    We refer to Yosida and Hewitt (1952), Rao (1958), and Peressini (1967).

  24. 24.

    Dubey and Mitra (2013) provide an example of a non-constructible relation on X that satisfies the Pigou-Dalton transfer principle while its representation has been established by Sakamoto (2012).

  25. 25.

    The map \(\liminf \) violates additivity: let \(x=(1,0,1,0,\ldots )\) and \(y=(0,1,0,1,\ldots )\), then \(\liminf (x)= \liminf (y)=0\) while \(\liminf (x+y)=1\). The map \(\liminf \), however, still fits in the Chichilnisky approach.

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Authors and Affiliations

  1. Center for Economic Studies, Katholieke Universiteit Leuven, Naamsestraat 69, 3000, Leuven, Belgium

    Luc Lauwers

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  1. Luc Lauwers
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  1. Columbia University, New York, New York, USA

    Graciela Chichilnisky

  2. Department of Socio-Economics, Vienna University of Economics and Business, Vienna, Austria

    Armon Rezai

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Lauwers, L. (2016). The Axiomatic Approach to the Ranking of Infinite Streams. In: Chichilnisky, G., Rezai, A. (eds) The Economics of the Global Environment. Studies in Economic Theory, vol 29. Springer, Cham. https://doi.org/10.1007/978-3-319-31943-8_12

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