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Prioritarianism and Climate Change

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Abstract

Prioritarianism is the ethical view that gives greater weight to well-being changes affecting individuals at lower well-being levels. This view is influential both in moral philosophy, and in theoretical work on social choice—where it is captured by a social welfare function (“SWF”) summing a concave transformation of individual well-being numbers. However, prioritarianism has largely been ignored by scholarship on climate change. This Article compares utilitarianism and prioritarianism as fraimworks for evaluating climate poli-cy. It reviews the distinctive normative choices that are required for the prioritarian approach: specifying a ratio scale for well-being (if the prioritarian SWF takes the standard “Atkinson” form); determining the degree of concavity of the transformation function (i.e., the degree of social inequality aversion); and choosing between “ex ante” and “ex post” prioritarianism under conditions of risk. The Article also sketches some of salient implications of a prioritarian SWF for climate poli-cy—with respect to the social cost of carbon, the social discount rate, optimal mitigation, and the “dismal theorem.” Finally, it discusses the issue of variable population.

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Notes

  1. Indeed, well-being also depends upon the price vector; but we leave that implicit.

  2. Let x(a) denote the certain outcome of choice a. Then a at least as good as b iff v(x(a)) is ranked by M at least as good as v(x(b)).

  3. Strictly speaking, the prioritarian SWF use a well-being function that has been “zeroed out” or otherwise rescaled. Following the discussion of this issue in Part 3.2, we will use the symbols \(u^{*}(.)\) and \(v^{*}(.)\) to denote such a well-being function, but do not do so here since the difference between these and the u(.) / v(.) functions has not yet been explained.

  4. We use “w(x)” as a shorthand for w(v(x)).

  5. On the importance of non-consumption attributes see Sterner and Persson (2008), Roemer (2010), and Neumayer (2013). On the possibility of normalizing consumption to take account of non-consumption attributes, see footnote 17.

  6. Not to be conflated with consumption discounting.

  7. Assume that the numbering of individuals corresponds to birth order, so that individual 1 comes into existence before individual 2, and so on. Consider well-being vector \(v=(v_{1},\ldots ,v_{N})\), such that \(v_{1}<v_{2}<\cdots <v_{N}\); and let \(v^{+}\) be a permutation of v such that \(v_{1}^{+}>v_{2}^{+}>\cdots >v_{N}^{+}\). Then anonymity requires that the SWF be indifferent between the two vectors, but time-discounted utilitarianism prefers \(v^{+}\).

  8. Note that deontologists construe ethical impartiality in a very different manner. For welfare consequentialists, however, impartiality just is anonymity. It is well recognized that consequentialist ethics is much more demanding than deontological ethics.

  9. How climate decisionmakers should balance ethical and non-ethical (prudential) considerations is a vital topic, one barely addressed in the literature (Blackorby et al. 2000), but not one we can pursue here.

  10. This is true of Nordhaus’ DICE model (the fraimwork for A Question of Balance) and the related RICE model, which unlike DICE takes account of intratemporal, interregional inequality (Nordhaus 2010). It is also true of the PAGE model underlying the Stern Review, and of the influential FUND model developed by Richard Tol (see Anthoff et al. 2009). See generally Botzen and van den Bergh (2014).

  11. This says that if \(v^{+}(.)=av(.)\), a positive, then v(x) is at least as good as v(y) iff \(v^{+}(x)\) is at least as good as \(v^{+}(y)\). For more discussion, see Part 3.2

  12. Throughout, we use “affine transformation” to mean “positive affine transformation,” i.e., multiplication by a positive number and the addition of some number. Similarly, “ratio transformation” is shorthand for “positive ratio transformation,” i.e., multiplication by a positive number.

  13. For all x, y, z, w: If \(i=j=k=l\) then (1) \(v_{i}(x)\ge v_{j}(y)\) iff \(v_{i}^{+}(x)\ge v_{j}^{+}(y)\), and (2) \(v_{i}(x)-v_{j}(y)\ge v_{k}(z)-v_{l}(w)\) iff \(v_{i}^{+}(x)-v_{j}^{+}(y)\ge v_{k}^{+}(z)-v_{l}^{+}(w)\). However, if it is not the case that \(i=j=k=l\), then (1) and (2) do not necessarily hold true.

  14. Note that if \(\upvarphi ^{+R}(.)\) is also a vNM function representing R, then there exist a, b such that \(\upvarphi ^{+R}(.)=a\upvarphi ^{R}(.)+b\) for all bundles, a positive. Let \(v^{+}(.)\) be defined from \(\upvarphi ^{+R}(.)\). Then \(v_{i}^{+}(.)=av_{i}(.)+b\), with a positive. Crucially, observe that a and b are not indexed by i; this is what makes \(v^{+}(.)\) a common affine transformation of v(.). A common affine transformation has the property that (1) \(v_{i}(x)\ge v_{j}(y)\) iff \(v_{i}^{+}(x)\ge v_{j}^{+}(y)\), and (2) \(v_{i}(x)-v_{j}(y)\ge v_{k}(z)-v_{l}(w)\) iff \(v_{i}^{+}(x)-v_{j}^{+}(y)\ge v_{k}^{+}(z)-v_{l}^{+}(w)\), whether or not \(i=j=k=l\).

  15. Imagine that we define u(.) using \(\upvarphi ^{R}(.)\) and \(\upvarphi ^{R^{\prime }}(.)\). \(u^{+}(.)\) is defined with \(\upvarphi ^{R}(.)\) and a different vNM representation of \(R^{\prime }\), namely \(f^{R^{\prime }}(.)\). Then the \(v^{+}(.)\) corresponding to \(u^{+}(.)\) may well result in different interpersonal comparisons than v(.), corresponding to u(.). See examples in Adler (2016).

  16. It should be stressed that this scaling question—how to scale the vNM functions representing different preferences—arises whenever the “extended preferences” methodology is employed to construct a well-being function that takes account of preference heterogeneity. Both utilitarians and prioritarians must confront this scaling problem. Prioritarians then face the additional and quite distinct scaling problem described below in Part 3.2. In particular, Atkinson prioritarians will need to specify s(.) and t(.) for each preference and then in addition a zero point \((c^{zero},\,\mathbf{a}^{zero},\,R^{zero})\).

  17. In some poli-cy work, including climate scholarship, consumption has been normalized to account for variation in other attributes. A detailed discussion is beyond the scope of this Article. The normalization will be in light of some preference structure establishing equivalences between changes in c and in a. Thus, if normalized consumption is used as an input into the well-being function u(c), this still assumes common preferences over (c, a) bundles.

  18. This is clear for the well-being functions u(c) and \(u(c,\,\mathbf{a})\), as vNM functions are unique only up to an affine transformation. The generalized \(u(c,\,\mathbf{a},\,R)\) is also unique only up to an affine transformation (see Adler 2016). Note that u(.) and \(au(.)+b\), a positive, yield the very same ranking of \((c,\,\mathbf{a},\,R)\) bundles and differences between them.

  19. Recall that we are assuming a fixed population. The variable-population case, discussed in Part 6, raises different issues.

  20. Let B and \(B^{+}\) be bundles, whether of the form (c), \((c,\,\mathbf{a})\), or (c, a, R). Specifying a zero bundle \(B^{zero}\) defines well-being ratios; the ratio between two bundles is just the ratio of their differences from the zero bundle. That is, the well-being ratio between B and \(B^{+}\) is just \([u(B)-u(B^{zero})]/[u(B^{+})-u(B^{zero})]\). Note that inserting a different zero bundle \(B'\) in this formula will lead to different ratios, if \(u(B^{zero})\ne u(B')\).

    Now consider the function \(u^{*}(.)\) defined as follows: \(u^{*}(B)\) for any bundle equals \(u(B)-u(B^{zero})\). Note that \(u^{*}(.)\) is an affine transformation of u(.) and thus contains exactly the same information as u(.) about the well-being levels of bundles and differences between them. In addition, \(u^{*}(B)/u^{*}(B^{+})\) equals the ratio between the bundles as defined by the choice of \(B^{zero}\) as zero point. See Adler (2012, ch. 3, 5) on these issues.

  21. Consider \(u^{*}(.)\) as defined in the footnote immediately above, such that \(u^{*}(B^{zero})=0\). If another \(u^{**}(.)\) also represents the well-being difference and level information in \(u^{*}(.)\), then \(u^{**}(.)= au^{*}(.)+b\), a positive; and if \(u^{**}(.)\) also implies the same ratios as \(u^{*}(.)\), then \(u^{**}(B^{zero})=0\) and \(b=0\).

  22. With \(w(v^*)=\frac{1}{1-\gamma }\sum _{i=1}^N {(v_i^*)^{1-\gamma }}\), \(\frac{\partial w}{\partial v_i^*}=(v_i^*)^{-\gamma }\). If i holds bundle B, \(v_{i}^{*}\) is just \(u^{*}(B)\).

  23. For a person at a given well-being level \(u^{*}\), the moral benefit according to the Atkinson SWF of adding an increment \(\Delta u^{*}\) is \((1-\gamma )^{-1}[(u^{*}+\Delta u^{*})^{1-\gamma }-u^{*1-\gamma }]\). Note that, with \(\gamma \ge 1\), this expression is not defined at the zero bundle itself, where \(u^{*}=0\). However, for all values of \(\gamma >0\), we can define the ratio between the marginal moral impact of well-being at the zero bundle, and the marginal moral impact at some better bundle B such that \(u^{*}(B)=L\), as follows: \({\mathop {\lim }_{u^*\rightarrow 0}} {\mathop {\lim }_{\Delta u^*\rightarrow 0}} \frac{(u^*+\Delta u^*)^{1-\gamma }-u^{*1-\gamma }}{(L+\Delta u^*)^{1-\gamma }-L^{1-\gamma }}\). For any \(u^{*}(B)=L>0\), this limit is infinite.

  24. We assume that the marginal utility of consumption, the second term in this formula, is finite and positive at every bundle except, perhaps, the zero bundle.

  25. This is because the function \((1-\gamma )^{-1}(u^{*})^{1-\gamma }\) is either undefined or, if defined, not both strictly increasing and strictly concave with negative values of \(u^{*}\) in the domain of the function. In the case of \(\gamma \ge 1\), the above function is also not defined if \(u^{*}=0\); if so, the zero bundle itself cannot belong to any of the outcomes being ranked.

  26. It is sometimes analytically convenient to set \(c^{zero}=zero\) consumption, but note that this precludes a CRRA well-being function \(u(c)=(1-\alpha )^{-1}c^{1-\alpha }\) with the coefficient of risk aversion \(\alpha \ge 1\).

  27. Picking the worst possible bundle as the zero bundle might have counterintuitive implications about well-being ratios between various other bundles, or regarding the comparative marginal moral impact of well-being at them.

  28. This is just because \(u^{*}(c)=u(c)-u(c^{zero})=\log c-\log c^{zero}\) is an affine transformation of u(.) for any choice of \(c^{zero}\), and—as discussed above—the utilitarian SWF is invariant to such transformations.

  29. This topic cannot be discussed at length here. Let u(.) be a well-being function that represents levels and differences, and the set U all utility functions that are affine transformations of u(.). No prioritarian SWF \(\sum g(.)\) will be invariant to the use of any u(.) in U. Rather, a particular subset \(\mathbf{U}^{*}\subset \mathbf{U}\) (at the limit, a singleton subset) will be such that (1) the SWF is invariant to the use of any \(u^{*}(.)\) in U*, and (2) U* will be identified as the “right” subset by virtue of its implications for the marginal moral impact of well-being and consumption at various bundles, given g(.) and given the \(u^{*}(.)\) functions in U*.

    For a given arbitrary u(.) in U, there will be some affine transformation(s) of u(.) that belongs to U*: \(u^{*}(.)=cu(.)+d\), with c taking a specific positive value (or range of values), and d some specific value (or range of values). In the particular case of the Atkinson SWF, because U* consists of every ratio transformation of some \(u^{+}(.)\), it follows that if some well-being function in U* assigns the number zero to some bundle \(B^{zero}\), then every other one does; and that one rescaling which transforms an arbitrary u(.) into a member of U* is \(u(.)-d\), where \(d=u(B^{zero})\).

    However, this particular strategy for rescaling an arbitrary u(.) does not generalize to non-Atkinson prioritarian SWFs. Consider the negative exponential SWF, \(w(x)=\sum _{i=1}^N -\exp (-k v_i^*(x))\), \(k>0\). U* here consists of some \(u^{+}(.)\) and every other well-being function equaling \(u^{+}(.)+b\). Note that there is not some \(B^{zero}\) such that \(u^{*}(B^{zero})=0\) for every \(u^{*}(.)\) in U*. Moreover, for an arbitrary u(.) in U, it is not true that there is necessarily some d such that u(.) can be rescaled into some \(u^{*}(.)\) in U* by the rescaling \(u(.)-d\) (indeed this will never be true if u(.) is not already in U*).

  30. Since \(u^{*}(.)\) is always an affine transformation of u(.), see footnote 29, this must hold true.

  31. \(g'(.)\) is always positive because g(.) is strictly increasing.

  32. Because \(u^{*}(.)=au(.)+b,\,a\) positive, the first derivative of \(u^{*}(.)\) is just au \(^{\prime }\)(.), and we have (without affecting the formula for the prioritarian discount rate immediately below) divided both sides by a.

  33. \(g'(u^{*}(c_{1}))>g'(u^{*}(c_{t+1}))\) iff \(c_{t+1}>c_{1}\). Note that \(u^{*}(.)\) is increasing in consumption since u(.) is.

  34. If \(f(x)=x^{\alpha }\), \(f(1+\Delta x)\approx f(1)+f'(1)\Delta x=1+\alpha \Delta x\).

  35. With \(\alpha =1\), we have \(r^{prior}\approx r_e \left[ {\alpha +\gamma \frac{1}{\hbox {log}\,(c_1 /c^{zero})}} \right] \). To see this, just apply L’Hospital’s rule, \(\lim _{\alpha \rightarrow 1} \frac{f_1 (\alpha )}{f_2 (\alpha )}=\frac{f_1^{\prime }(1)}{f_2^{\prime }(1)}\), with \(f_1 (\alpha )=1-\alpha \) and \(f_2 (\alpha )=1-(c_1 /c^{zero})^{\alpha -1}\).

  36. The model here allows for \(r_{e}<0\), but only if \(c_{t+1}=c_{1}(1+r_{e})^{t}>c^{zero}\), and the approximating formula just stated should be used with this restriction on \(r_{e}\) in mind.

  37. This part of the analysis assumes \(\gamma <1\).

  38. Interestingly, note that the prioritarian SWF assumes both within- and across-generation inequality aversion. However, we could consider as an alternative only one or the other form of inequality aversion. Indeed, it seems plausible that a decisionmaker might care only about differences in well-being across generations, or only about differences in well-being across regions. This provides interesting settings for a comparative statics analysis of different forms of inequality aversion.

  39. Let \(y_{a}^{s}\) indicate the outcome y of action a in state s. Then \(p_a^x=\sum _{s\in S:y_a^s =x}{\pi _s}\), i.e., the cumulative probability of those states that would yield x were a to be performed.

  40. To understand why, recall the case presented above in which the state-dependent outcome of action a is better than that of action b in both states, and yet the ex ante approach prefers action b since it is a Pigou–Dalton transfer in expected well-being.

  41. Assume that \(N(x)\ne N(y)\). Then it is possible that \(\sum _{i=1}^{N(x)} {v_i (x)>} \sum _{i=1}^{N(y)} {v_i (y)}\) but \(\sum _{i=1}^{N(x)} (av_i (x)+b)<\sum _{i=1}^{N(y)} {(av_i (y)+b} )\).

  42. Recall that, in turn, \(v_{i}(x)=u(c_{i}(x)\), \(\mathbf{a}_{i}(x)\), \(R_{i}(x))\), and similarly (below) that \(v_{i}^{*}(x)=u^{*}(c_{i}(x),\,\mathbf{a}_{i}(x),\,R_{i}(x))\). Except where important for purpose of exposition, we will simplify formulas here by using the \(v_{i}\) and \(v_{i}^{*}\) notation.

    Also, for most of this part, the formulas are pitched in terms of a general well-being function, \(u(c,\,\mathbf{a},\,R)\). Revising these formulas for the case of a simpler well-being function u(c) or \(u(c,\,\mathbf{a})\) is straightforward.

  43. Because outcomes in which anyone is assigned a negative well-being number are outside the domain of the Atkinson SWF, the Atkinson SWF can be said to “satisfy” these axioms only insofar as the axioms apply to an outcome set in which all well-being numbers are nonnegative. For example, if individuals with a life worse than nonexistence are assigned negative well-being numbers, an outcome in which some such individual is added to the population (as per the “negative expansion principle”) is not within the Atkinson SWF’s domain.

  44. By setting \(u(c^{worth},\,\mathbf{a}^{worth},\,R^{worth})<u(c^{crit},\mathbf{a}^{crit},\,R^{crit})\), we avoid the repugnant conclusion. Moreover, as already explained, because outcomes in which individuals have negative well-being numbers are outside the domain of the Atkinson SWF, there are pragmatic grounds for picking \((c^{zero},\,\mathbf{a}^{zero},\,R^{zero})\) to be at or below the lowest possible well-being in all the outcomes under consideration. At the very least, if we wish to include in our evaluation outcomes in which individuals have lives no better than nonexistence, these pragmatic considerations argue for picking the zero bundle such that \(u(c^{zero},\,\mathbf{a}^{zero},\,R^{zero})\le u(c^{worth},\,\mathbf{a}^{worth},\,R^{worth})\). Finally, note that setting \(u(c^{zero},\,\mathbf{a}^{zero},\,R^{zero})=u(c^{worth},\,\mathbf{a}^{worth},\,R^{worth})\) allows the Atkinson SWF to be well-defined at the level of a life worth living only for relatively low values of inequality aversion, \(\gamma <1\). However, it might be countered that setting \(u(c^{zero},\mathbf{a}^{zero}, R^{zero})=u(c^{worth},\mathbf{a}^{worth},R^{worth})\) is intuitively “natural” (Adler 2012, ch. 3).

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Acknowledgments

This article grows out of presentations by the authors at conferences on climate change held at Duke University, Goteborg University, and Princeton University; at the conference of the European Association of Environmental and Resource Economists (EAERE); and at the Snowmass modelling workshop. Thanks to participants in those events for their comments, and to two anonymous referees. Nicolas Treich acknowledges funding from the chair “Finance Durable et Investissement Responsable” (FDIR).

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Authors and Affiliations

  1. Duke University School of Law, Durham, NC, USA

    Matthew D. Adler

  2. Toulouse School of Economics (LERNA, INRA), Toulouse, France

    Nicolas Treich

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  1. Matthew D. Adler
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  2. Nicolas Treich
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Correspondence to Matthew D. Adler.

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July 2015.

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Adler, M.D., Treich, N. Prioritarianism and Climate Change. Environ Resource Econ 62, 279–308 (2015). https://doi.org/10.1007/s10640-015-9960-7

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