7.1. Arguments for a Zero Rate of Pure Time Preference

The basic ‘argument’ for a zero rate of pure time preference is from impartiality. Accepting a zero rate of pure time preference merely amounts to treating utility as equally valuable, regardless of when it occurs. But of course (runs the thought) the value of utility is independent of temporal location: there is no possible justification for holding that the value of (say) curing someone’s headache, holding fixed her psychology, circumstances and deservingness, depends upon which year it is. The axiomatic nature of impartiality is endorsed both by many of the seminal articles on the subject (Sidgwick Reference Sidgwick1890; Ramsey Reference Ramsey1928; Pigou Reference Pigou1932; Harrod Reference Harrod1948; Solow Reference Solow1974) and by many current authors (Cline Reference Cline1992; Broome Reference Broome2008; Dasgupta Reference Dasgupta2008; Dietz et al. Reference Dietz, Hepburn, Stern, Basu and Kanbur2008; Buchholz and Schumacher Reference Buchholz and Schumacher2010; Gollier Reference Gollier2013).

To this we may add two further arguments. The second argument proceeds from the Pareto principle, and points out that this principle is inconsistent with a non-zero rate of pure time preference (Cowen Reference Cowen, Parfit, Laslett and Fishkin1992). To see the inconsistency, suppose that a particular person – Sarah, say – could live either in this century or the next. Consider two states of affairs that differ over when Sarah lives. Suppose that Sarah’s well-being is slightly better in the state of affairs in which she lives later, while everyone else’s well-being is unchanged. Then according to the Pareto principle, the ‘Sarah lives later’ state of affairs is better. But according to a value function whose rate of pure time preference is positive, this state of affairs may be worse. Thus δ ≠ 0 is inconsistent with the Pareto principle.

Third, if δ > 0 for negative (i.e. past) as well as positive (future) times, we have the absurd implication that deaths in the past were worse than deaths now. If, on the other hand, δ > 0 only for positive times, then we have temporal relativity, and this temporal relativity tends to lead to temporal inconsistency in judgements. We pause to explain the latter point (see also Broome Reference Broome2004: sec. 4.3; Reference Broome2012: pp.148–52).

Let us define a schedule of discount factors to be an assignment of a discount factor to each pair consisting of the time of the benefit to be evaluated and the time at which the evaluation is carried out: thus Rⁱ _j is the discount factor used by an evaluator at time t_i when evaluating costs or benefits occurring at time t_j . By definition, a schedule of discount factors is time-neutral iff the ratios $\frac{{R^i_j}}{\scriptsize{R^i_{j^\prime }}}$ are independent of i; otherwise it is time-relative. The point is then that if $\frac{\scriptsize{R^i_j}}{\scriptsize{R^i_{j^\prime }}} \ne \frac{\scriptsize{R^{i+1}_j}}{\scriptsize{R^{i+1}_{j^\prime }}}$ , then evaluations carried out at times i and i + 1 will disagree with one another on (some) decisions that involve trading off well-being at time j with well-being at time j′, even in the absence of any new information. Furthermore, the agents doing the evaluating at time i will be in a position to foresee that they will change their minds at i + 1, despite not having learnt anything new. This sort of time-inconsistency is generally taken to be an indicator of irrationality.

In the present case: Suppose that the rate of pure time preference, at any evaluation time t, is zero for all times prior to t, and some positive constant δ > 0 for all times later than t. Let t ₁, t ₂ be any pair of times with t ₂ > t ₁. We consider an operation that sacrifices c ₁ units of utility at time t ₁ in return for an increase of b ₂ units of utility at time t ₂. We evaluate this operation, first from the point of view of t ₁, and then from the point of view of t ₂. An evaluator at t ₁ applies a discount factor of exp(− δ(t ₂ − t ₁)) to utility gains occurring at the later time t ₂, and therefore deems the operation to be an overall improvement iff nexp(− δ(t ₂ − t ₁)) > m. The evaluator at t ₂, however, judges one unit of utility to be equally valuable whether it occurs at t ₁ or t ₂, and hence favours our operation iff n > m. This second criterion is of course less stringent than the first; there are thus operations such that, by the lights of his own evaluation strategy, an actor at t ₁ should turn them down, but the same actor at t ₂ should regret having done so.

Suppose, for example, that Ben gets the same amount of utility from each doughnut he eats, regardless of which day he eats it on, and (at least for small numbers of doughnuts) regardless of how many other doughnuts he has eaten on the same day. But suppose that each day, Ben discounts his own future utility in the manner suggested, while regarding all present and past utility as equally valuable: specifically, suppose that he discounts utility at the (extreme) rate of 60% per day. On Monday, he is offered a choice between (option 1) eating one doughnut on Tuesday or (option 2) eating two on Wednesday; given his Monday values, he judges option 1 to be better. He still agrees with this judgement on Tuesday. But come Wednesday, he has changed his mind about the relative value of Tuesday doughnuts and Wednesday doughnuts: he now thinks that Tuesday doughnuts and Wednesday doughnuts are equally valuable, and thus judges option 2 to be better than option 1. It is too late, of course: he cannot now do anything about it, but he regrets the decision he made on Monday. Furthermore, when deliberating on Monday, he was already in a position to foresee that he would thus later regret the decision he was making.

It is worth noting well, however, that the only sort of inconsistency that can result from the discounting structure in question is the phenomenon of foreseeable regret that we have seen in this example. This is arguably not as bad as foreseeable backtracking, in which an agent decides at time i to pursue a certain course of action at time i + 2, while recognizing that at time i + 1 she will no longer think that this course of action is best, and so will attempt to reverse her decision. The latter but not the former raises issues of commitment (‘tying oneself to the mast’); cf. the discussion of declining discount rates in Section 9.2.

7.2. Arguments for a Positive Rate of Pure Time Preference

Most theorists see some significant prima facie force to (in particular) the intuition of impartiality reported above, but it is far from agreed that that intuition carries the day. We now consider four arguments for a positive rate of pure time preference.

Our first argument is a direct response to the impartiality intuition: that full impartiality, while perhaps a moral ideal, is not a requirement of morality. For example, according to common sense morality, it is at least permissible to give the interests of one’s friends and family more weight than those of strangers in one’s decisions; according to defenders of a positive rate of pure time preference, the latter is merely a manifestation of this permissible partiality in the intertemporal case (Arrow Reference Arrow, Portney and Weyant1999; Flanigan n.d.). In reply, several authors (Cowen and Parfit Reference Cowen, Parfit, Laslett and Fishkin1992; Caney Reference Caney2008; Beckstead Reference Beckstead2013) point out that to say that future utility is just as valuable as present utility is not to commit to a moral obligation to maximize overall value, and that Arrow’s objection seems to be more to the latter account of obligation (‘maximizing consequentialism’). To this, the proponent of the ‘permissible partiality’ argument might reply that his ‘social welfare function’, contrary to the interpretation we dubbed ‘natural’ in section 1, is not intended to be a representation of ‘overall value’ in the impartial sense; it is, rather, the function such that in his view the course of action that maximizes that function is the most advisable one, subject to the constraints of morality. The question then is whether this representation by a single discounted function is a sensible form for the non-maximizing-consequentialist view in question (as opposed, say, to the use of a fully impartial value function supplemented by principles expressing agent-centred prerogatives); the critics argue that it is not.

Second, and relatedly, many theorists hold that the use of a zero rate of pure time preference requires excessive sacrifice on the part of the present generation for the benefit of future generations (again, see Arrow Reference Arrow, Portney and Weyant1999). The relevant theory here concerns the optimal rate of saving. For example, Arrow calculates that under certain plausible-looking circumstances, a zero rate of pure time preference would require us to save over two thirds of our income for the benefit of future people. (This ‘excessive sacrifice’ argument is of course closely related to the ‘permissible partiality’ argument just stated; what the ‘excessive sacrifice’ argument adds is that impartiality (is not only not required, but also) is not a sensible option.)

It is instructive to compare this ‘excessive sacrifice’ argument in the context of intertemporal ethics with one that often crops up in the assessment of utilitarianism as a moral theory for the intratemporal case. According to utilitarianism, rich people (like ourselves) should give away the vast majority of our wealth to the desperately poor. Many (e.g. Scheffler Reference Scheffler1982) object that this implication is too demanding, and conclude that utilitarianism is false. The difference in the intertemporal case is that the fully impartial value function asks us to make these large sacrifices for people who are already richer than us. Thus, in practice the ‘excessive sacrifice’ argument is more compelling in the intertemporal case than in the intratemporal case. (The difference arises because of the possibility of generating a larger benefit in consumption terms for others than one’s own sacrifice, thanks to the phenomenon of investment. This effect has no analogue in the intratemporal case.)

Whether or not a value function embodying a zero rate of pure time preference does mandate an intuitively excessive level of saving, even under the assumption of a moral requirement to maximize, of course depends on the instantaneous utility function it is based on. Instead of concluding that δ > 0, some authors (for example, Asheim and Buchholz Reference Asheim and Buchholz2003; Dasgupta Reference Dasgupta2008) conclude that the utility function must be more concave than those considered by Arrow. Others (e.g. Dietz et al. Reference Dietz, Hepburn, Stern, Basu and Kanbur2008: 382) note that increasing the amount of assumed technical progress will also tend to decrease the optimal level of savings, for fixed δ. Another possibility is that (δ = 0 but) technical progress will push up growth to the point at which (for ordinarily assumed values of η) optimal savings rates are not excessive.

Third, there is a cluster of worries about adverse implications of time-impartiality in cases where the consumption or utility streams under consideration extend into the infinite future. The basic worry can be seen by considering the natural value function for infinite utility streams in the absence of discounting, ∑^∞ _{i = 0} u_i . Even if there is a common maximum value for each instantaneous utility u_i , the sum ∑^∞ _{i = 0} u_i has no upper bound, and (in addition) for infinitely many possible utility streams the sum will fail to converge at all. It follows that no utility stream is maximal relative to the preference order over utility streams that is represented by this value function, and (in addition) that the preference ‘order’ thereby represented is incomplete (strictly speaking, it is a preorder rather than an order on utility streams).

Developing this basic observation, an extensive literature investigates the consistency of time-impartiality with various other apparently plausible constraints on preference orderings in axiomatic frameworks (as opposed to: in the context of considering particular value functions). Inconsistency results are established by e.g. Koopmans (Reference Koopmans1960), Diamond (Reference Diamond1965) and Epstein (Reference Epstein1986).

Many authors take these concerns relating to infinite contexts to supply sufficient motivation for adopting a positive rate of pure time preference (note that with bounded instantaneous utilities, the discounted sum ∑^∞ _{i = 0}β ⁱ u_i (β < 1) does converge for every possible infinite utility stream (u_i )). Alternative responses include the following. Gollier (Reference Gollier2013: 32) questions the motivation for insisting on the existence of an optimal consumption stream, since one can determine whether or not an alternative is an improvement over the status quo even if neither status quo nor alternative are optimal; this does not, however, address the second worry that the undiscounted sum may simply fail to converge for either or both alternatives. Svensson (Reference Svensson1980) questions the motivation for the ‘continuity’ assumptions involved in many of the impossibility results, and establishes the consistency of a complete preference ordering and time-impartiality with the remainder of the Koopmans-Diamond axioms (see also Broome Reference Broome1992: 140–5). A third response is that the utility streams under consideration in real scenarios do not in fact involve an infinite time horizon, since the human race will eventually become extinct due to e.g. the heat death of the Universe; if so, then (the response continues) it is perverse to base ethics for the actual world on paradoxes of infinity that one would have to face up to in distant alternative possible worlds. From the point of view of practical applications (if not of fundamental principle), perhaps the most powerful response is that of Dasgupta (Reference Dasgupta2008: 157): even if the technical considerations in question do induce one to adopt a positive rate of pure time preference, an arbitrarily small positive value will suffice. The theorems, for instance, would supply no grounds for any objection to δ = 10⁻¹⁰⁰% per annum.

A fourth argument for a positive rate of pure time preference proceeds from the premise that the actions of a government should be selected on the basis of aggregating the preferences of present members of the body politic. The argument starts from the empirical premise that people do in fact discount future utility (the empirical literature on this issue is surveyed in (Frederick et al. Reference Frederick, Loewenstein and O'Donoghue2002)). Therefore, the argument runs, the government should respect this preference, and itself discount future utility when evaluating projects. Since a close analogue of this last argument for the discount rate on goods has cropped up in the discussion of climate change, we delay discussion of its soundness until Section 10.

8.1. Three Roles for η

A higher value of η corresponds to a more concave utility function. In the standard approach – that is, the approach that seeks to maximize the expectation value of the value function (4) – this leads η to play three conceptually distinct roles, related to the degree of aversion to (respectively) risk, intratemporal inequality and intertemporal inequality. We pause to explicate these three roles in turn.

On risk aversion: The expectation value E[X] of a random variable X is the probability-weighted average of the possible values of X. We then say that a value function exhibits risk aversion iff it ranks a given lottery (in the sense of: an assignment of probabilities to consumption streams) as less good than a guarantee of the consumption stream that is the expectation value of those involved in the lottery: if, for example, it ranks a sure prospect of $100 as better than a 50:50 lottery yielding either $50 or $150. Assuming (as is standard; but see Section 9.3) the expected-value approach to uncertainty, this is the condition that E[V(c)] ⩽ V(E[c]). We rewrite (4) as

(8) $$\begin{equation} V = \int dt \cdot V_t, \end{equation}$$

where V_t = Δ(t) · N(t) · u(c(t)). Since E[V] = ∫dt · E[V_t ], we can abstract from intertemporal issues, and consider a representative time t; we will have risk-aversion in the above sense if the utility function is such that for any lottery over consumption level c, E[u(c)] ⩽ u(E[c]). A standard argument (Eeckhoudt et al. Reference Eeckhoudt, Gollier and Schlesinger1995: 8) establishes that this condition obtains if and only if the utility function is concave (that is, while u′ > 0, u′′ < 0).

This tells us when a utility function exhibits risk aversion, but not yet when one utility function exhibits more risk aversion than another. That is, we have not yet defined a notion of degree of risk aversion. To do so, we define the risk premium associated with a given lottery to be the amount of consumption π such that enjoying a sure consumption level that is equal to the expected consumption level in the lottery less π is ex ante equivalent to facing the lottery: that is, the quantity π such that E[u(c)] = u(E[c] − π). We then say that utility function u ₁ exhibits more risk aversion than utility function u ₂ if and only if the risk premium for every lottery is higher according to u ₁ than according to u ₂. It is straightforward to show (Pratt Reference Pratt1964) that one utility function exhibits higher risk aversion than another in this sense if and only if the first yields a higher value of η at all consumption levels than the second. (Strictly: If the first has η everywhere at least as high as the second, and strictly higher for at least one consumption level in every interval. See (Pratt Reference Pratt1964: Thm. 1).)

On aversion to inequality: As a first pass (but see below), we say that a value function exhibits aversion to inequality if, given two individuals with unequal consumption levels, a transfer of a certain amount of consumption from the higher-consuming to the lower-consuming individual (which transfer, however, does not reverse the ordering of the two individuals’ levels) always increases value. (This is the ‘Pigou-Dalton principle’ (Sen Reference Sen1973; Fleurbaey Reference Fleurbaey and Zuber2012; Adler Reference Adler2013).) There are several possible rationales for aversion to inequality. First, consumption exhibits diminishing marginal utility, so that a given increment in consumption makes less difference to one’s welfare the better off one (already) is. Second, the degree of welfare concomitant on a given level of consumption depends how that consumption level compares to the consumption levels of others with whom one is in contact, both for purely psychological reasons and due to social exclusion effects (Payne Reference Payne2000; Alesina et al. Reference Alesina, Tella and MacCulloch2004; Di Tella and MacCulloch Reference Di Tella and MacCulloch2006; Dynan and Ravina Reference Dynan and Ravina2007; Morgan et al. Reference Morgan, Burns, Fitzpatrick, Pinfold and Priebe2007; Clark et al. Reference Clark, Frijters and Shields2008; Oishi et al. Reference Oishi, Kesebir and Diener2011). Third, some take inequality of resources simply to be intrinsically bad, perhaps for reasons of fairness. Finally, some theorists believe that inequality even of welfare levels is intrinsically a bad thing (Temkin Reference Temkin1993), so that inequality of consumption levels is bad to a degree beyond that that would be mediated via by any of the above routes. To probe the relation between inequality aversion and η, we consider the cases of intra- and intertemporal inequality separately.

On aversion to intratemporal inequality: As we noted in Section 3, since the value function (4) considers only matters of average per capita consumption at a given time, that value function altogether ignores issues of intratemporal inequality. More fundamentally, we would want to consider instead the value function (3):

$$\begin{equation*} V_3 = \int dt \cdot \Delta (t) \cdot \sum _{i=1}^{N(t)} u \left( c_i(t) \right), \end{equation*}$$

where i indexes individual people, and c_i (t) is individual i’s consumption at time t. We say that the value function is averse to intratemporal inequality if, given any two individuals with differing consumption levels at a given time t, overall value would be increased by giving each of the two individuals a consumption level that is half of their combined consumption, and leaving all others’ consumption levels unaffected. An analysis structurally identical to that above then shows that V ₃ is averse to intratemporal inequality if and only if u is concave, and that one value function of the form V ₃ is ‘more averse to intratemporal inequality’ than another if and only if the first arises from a utility function u that everywhere has a higher value of η than the second.

Similar considerations apply to the analysis of intertemporal inequality, that is, inequality in consumption levels occurring at different times. To see this, return to the simplified value function (4), and consider a case in which two individuals, living at different times, enjoy different levels of consumption c(t ₁), c(t ₂). The analysis in the intertemporal case is slightly complicated by the presence of the weighting factors Δ(t), but we can deal with the complications by defining the weighted average $\skew2\tilde{E}(c(t_1), c(t_2)) := \Delta (t_1)c(t_1) + \Delta (t_2) c(t_2)$ . If we say that (4) is averse to intertemporal inequality just in case its value is always increased by replacing both consumption levels in the unequal intertemporal pair with their weighted average $\tilde{E}(c(t_1), c(t_2))$ , then the same analysis as above establishes that (4) is averse to intertemporal inequality iff u is concave, and a similar extension of the reasoning establishes that one value function of the form (4) is ‘more averse to intertemporal inequality’ than another if and only if the first arises from a utility function u that everywhere has a higher value of η than the second.

It is this third role of η that corresponds to its importance in the debate over the discount rate. If (as is standardly assumed) future generations will, climate change or no climate change, enjoy higher levels of average per capita consumption than present generations – if, that is, consumption growth g is positive – then a higher value of η corresponds to a higher discount rate. (If, instead, future generations are worse off than the present generation – due to climate change or otherwise – then of course the relationship between η and the discount rate is reversed.)

8.2. Models that Separate the Three Roles

Several authors (e.g. Dietz et al. Reference Dietz, Hepburn, Stern, Basu and Kanbur2008; see also Selden Reference Selden1978; Epstein and Zin Reference Epstein and Zin1989) have urged that since the three roles that are played by η in the standard approach are conceptually distinct, to represent all three by a single parameter is both to rule out without justification the possibility that the parameters relevant to the three roles vary independently of one another, and to court unnecessary confusion.

On the former point: it does not seem incoherent (at least) to exhibit relatively little aversion to risk, but great aversion to any form of interpersonal inequality (intra- or inter-temporal) (Beckerman and Hepburn Reference Hepburn and Groom2007). Still less does it seem incoherent to exhibit great aversion to intratemporal inequality, but relatively little aversion to intertemporal inequality, if the reason for inequality aversion is based on the psychological or social exclusion effects mentioned in Section 8.1, as opposed to any view that inequality either of resources or of welfare is simply intrinsically bad. One might therefore seek a model that allows parameters for the three types of aversion to be varied independently, so that the resulting positions can be represented within the terms of the model.

On the latter point: As we noted at the outset, discussions of climate change often note that the most significant harms due to climate change will occur in the relatively far future, while the costs of any mitigative action will be incurred much sooner. Assuming that consumption growth is positive, this fact corresponds to a prima facie tendency for increasing η to favour less aggressive mitigative action; this is the tendency that is captured in the contribution of η to the discount rate in the basic Ramsey equation (6). But that basic Ramsey model ignores both intratemporal inequality and risk; a full analysis would carry out a line of the reasoning parallel to that of Section 5, but starting from the expectation value of the more detailed value function (3) rather than from the simplification (4). Notwithstanding the point that expected average per capita consumption is higher in the future than it is for those who would stand to bear the costs of mitigation, in such a fuller analysis, increasing η would place greater evaluative emphasis on incremental changes to climate impacts occurring in those future regions of the world and/or possible scenarios in which the potential beneficiaries of mitigation are actually poorer than the mitigators. Thus the relationships of η to intratemporal inequality and to risk potentially work against its relationship to intertemporal inequality, when we ask whether, overall, increasing η would favour more or less mitigative action. (See also the discussions in Dietz et al. Reference Dietz, Hope and Patmore2007, Reference Dietz, Hepburn, Stern, Basu and Kanbur2008: 378).

To say that the three notions are conceptually distinct, however, is not to say that there cannot be close links between them – perhaps close enough that no model of rational decision-making can vary them independently. Arguments to this effect concerning the relationship between risk aversion on the one hand and (either type of) inequality aversion on the other are given by Harsanyi (Reference Harsanyi1953, Reference Harsanyi1955, Reference Harsanyi1977).

8.3. Determining the Value of η

Setting aside the discussion of Section 8.2 and assuming the standard model, what then should the value of η be?

Many empirical studies aim to establish the extent to which real people do exhibit aversion to risk and interpersonal inequality in their choice behaviour. For the purposes of answering our normative question, it is important not to overstate the relevance of such empirical data: our question is what value for η our analysis should use, and it does not of course immediately follow, from the fact that people do employ a certain value for η, that they, we or anyone else should employ that value (Buchholz and Schumacher Reference Buchholz and Schumacher2010: 378). (It does not immediately follow; but that is not to say that it does not follow at all. Some authors would defend the inference using a ‘democracy argument’ that is a close cousin of the one discussed in Section 10.) That said, the empirical data may be informative nonetheless: a normative theorist advocating an η value that lies outside the range that empirical studies suggest that people normally use should be aware that he is so deviating from the statistical norm, and that he may be committed to passing a negative judgement on the values of the subjects in the empirical studies in question. We will first survey the more directly normative arguments, and then provide a brief survey of the empirical literature.

On the more directly normative side: Okun (Reference Okun1975: 91–5) outlines a ‘leaky bucket’ thought experiment, in which one envisages a transfer of resources from a richer to a poorer person (at specified initial wealth levels), and one asks how ‘lossy’ the transfer can be (that is, by how much the amount gained by the poorer person can fall short of the amount taken from the richer person) before the transfer ceases to constitute an overall improvement. If, for example, one judges that one person having £100k and another having £20k is neither better nor worse overall than one person having £60k and another having £30k (that is, the result of transferring £40k away from the better-off person but with a loss of 75% of the transferred resources), then in the context of the utilitarian model one is implicitly committed to an η value of about 1.2.Footnote ³ Taking another approach, Buchholz and Schumacher (Reference Buchholz and Schumacher2010) show that axiomatic conditions intended to capture aspects of ‘circumstance solidarity’ or ‘absence of envy’ variously require η = 1, η > 1, η = 2 and η = ∞; as witness the variety of these conclusions, however, the motivation for the axioms is open to serious question. Another approach considers the optimal rate of savings: as discussed in section 7, if one has an independent grip on the optimal balance between consumption and saving, then ‘working backwards’ can provide guidance on the correct value of η.

Turning now to the empirical studies: corresponding to the connections of η to both risk aversion and inequality aversion discussed above, data on either of the latter can be used to infer the values of η that experimental subjects are committed to, in their verbal or other behaviour. The ‘experimental subjects’ in these studies are sometimes individuals, sometimes governments. Focussing on risk aversion, Gollier (Reference Gollier2006) infers η ∈ [2, 4] from studies of individuals’ insurance behaviour. Carlsson et al. (Reference Carlsson, Daruvala and Johansson-Stenman2005) use a questionnaire involving hypothetical decision-making on behalf of grandchildren to probe individuals’ attitudes to risk aversion and inequality aversion separately; their results suggest a median η of 2–3 from the risk aversion study, and approximately 2 from the inequality aversion study.

Approaches based on government behaviour often focus on tax policy. If one assumes that the government accepts a ‘principle of equal sacrifice’ – that taxation is to be organized in such a way that each taxpayer forfeits (say) the same amount of utility by his tax contribution – then one can infer an implied value of η from the variation in tax level with income. Using this approach, Stern (Reference Stern, Artis and Nobay1977) argues that UK tax policy in 1973–4 implied an η-value of 1.97; Cowell and Gardiner (Reference Cowell and K. Gardiner1999) note that the analogous data from the late-1990s would imply η in the range 1.3–1.4.

Aside from the above-mentioned worries about the relevance of empirical data to normative questions, the use of these studies to inform a choice of η for policy purposes has been criticized on the grounds that the results of many of the studies are highly sensitive to an ‘arbitrary’ choice of sacrifice principle (Creedy Reference Creedy2006: 15), and on the ground that they illicitly assume that the value of η is independent of consumption level (Atkinson and Brandolini Reference Atkinson and Brandolini2010: 16). It is clear, however, that the more directly normative arguments surveyed above are to a large extent open to the same criticisms.

9.1. Weitzman’s Argument

In a seminal article, Weitzman (Reference Weitzman1998) claimed that the correct results are given by using an effective discount factor for any given time t that is the probability-weighted average of the various possible values for the true discount factor R(t): R _eff(t) = E[R(t)]. From this premise, it is easy to deduce, given the exponential relationship between discount rates and discount factors, that if the various possible true discount rates are constant, the effective discount rate declines over time, tending to its lowest possible value in the limit t → ∞. Weitzman (Reference Weitzman2001) uses this analysis to calculate a value for the effective discount rate as a function of time, using as input data the opinions of 2,160 economists on the ‘true’ value of the (constant) discount rate.

Weitzman did not, however, supply any fundamental justification for his assumption that the effective discount factor is the expectation value of the true discount factor. That assumption is equivalent to the claim that a project should be adopted iff its expected net present value (ENPV) is positive. Gollier (Reference Gollier2004) pointed out that if one adopts instead a criterion of positive expected net future value (ENFV), one obtains precisely the opposite of Weitzman’s result: the effective discount rate increases over time, tending as t → ∞ to its maximum possible value. Since neither criterion is obviously inferior to the other, the difference in results generated the so-called ‘Gollier-Weitzman puzzle’.

To make progress on this puzzle, we note that more fundamentally, one wants to start from expected utility theory: under conditions of uncertainty, a given project should be adopted if and only if its adoption increases expected value, where the ‘value’ in question aggregates over time as in equation (4). In general, this condition entails neither the ENPV nor the ENFV criterion. (To see this, consider a project involving a reduction in consumption − B ₀ at time 0, in return for a corresponding increase B_t at time t. Such a project increases expected value iff $B_t E [ \scriptsize{\frac{\delta V}{\delta C_t}}] > B_0 E [ \scriptsize{\frac{\delta V}{\delta C_0}} ]$ , where $\frac{\delta V}{\delta C_t}$ (respectively ( $\frac{\delta V}{\delta C_0}$ )), the functional derivative of value with respect to consumption at time t (resp. at time 0), measures the marginal rate at which overall value V increases per unit increase in consumption at time t (resp. 0), holding fixed consumption levels at other times. The project has positive ENPV, on the other hand, iff $B_t E [ \scriptsize{\frac{\frac{\delta V}{\delta C_t}}{\frac{\delta V}{\delta C_0}}}] > B_0$ ; since the random variables $\frac{\delta V}{\delta C_t}$ and $\frac{\delta V}{\delta C_0}$ are not in general independent, these criteria do not in general coincide. Similar remarks apply to the ENFV criterion.) The question, then, is whether the criterion of maximization of expected utility entails anything like Weitzman’s result given certain auxiliary assumptions, or in certain models. In the lively post-2004 debate over the Gollier-Weitzman puzzle (Hepburn and Groom Reference Hepburn and Groom2007; Buchholz and Schumacher Reference Buchholz and Schumacher2008; Gollier and Weitzman Reference Gollier and Weitzman2010; Freeman Reference Freeman2010; Gollier Reference Gollier and Weitzman2010; Traeger Reference Traeger2012a; Arrow et al. Reference Arrow, Cropper, Gollier, Groom, Heal and Newell2014), there is a widespread consensus that something like Weitzman’s original conclusion is correct, although participants to the debate continue to differ significantly over the reasons for this conclusion and the precise conditions under which it holds.

A recent literature in moral philosophy (Lockhart Reference Lockhart2000; Ross Reference Ross2006; Harman Reference Harman2011; Sepielli Reference Sepielli2013, Reference Sepielli2014; Weatherson Reference Weatherson2014; Nissan-Rozen Reference Nissan-Rozen2015; Gustafsson and Torpman Reference Gustafsson and Torpman2014; Mason Reference Mason2015) explores the possibility that evaluative, as well as empirical, uncertainty should be treated by means of expected utility (or ‘expected value’) theory. While this view is far from uncontroversial, the main sources of dissent stem either from the view that evaluative uncertainty is simply normatively irrelevant, or from the view that under evaluative uncertainty one should use the most-likely evaluative hypothesis, rather than proposing any alternative and non-trivial way to deal with such uncertainty. In the present context, the expected value approach suggests treating the values of δ and η, as well as g, as parameters that vary from one ‘state of nature’ to another. To the author’s knowledge, this approach has not been explored in any detail in the literature on discounting. Weitzman’s original (Reference Weitzman1998) argument was conducted at a level of generality sufficient to ensure neutrality between empirical and evaluative uncertainty, and his (Reference Weitzman2001) analysis fairly explicitly averages over evaluative and empirical uncertainty in the same way. Much of the later literature, investigating more detailed models in an attempt to justify one or another approach, tends (however) to restrict the scope of the uncertainty to the empirical. Arrow et al. (Reference Arrow, Cropper, Gollier, Groom, Heal and Newell2014) explicitly claim that normative and empirical uncertainty ‘require different approaches’ (Reference Arrow, Cropper, Gollier, Groom, Heal and Newell2014: 378); however, they neither offer any argument for this claim, nor propose any particular alternative treatment of evaluative uncertainty.

9.2. Declining Discount Rates

Even before Weitzman’s result, many theorists had suggested that the discount rate appropriate for application to the very far future was lower than that appropriate to the shorter term. The motivations for the suggestion are various (and dependent in part on the authors’ views on the methodological controversies discussed in this article). Those who take interest rates available on the credit markets as a reliable guide to discount rates observe that the interest rates available on government bonds, for instance decline as the period of the bond increases (Newell and Pizer Reference Newell and Pizer2003; Gollier et al. Reference Gollier, Koundouri and Pantelidis2008). Those who are willing to apply data on individual behaviour cite studies suggesting that individuals employ declining discount rates in their private decision-making (Loewenstein and Prelec Reference Loewenstein and Prelec1992; Henderson and Bateman Reference Henderson and Bateman1995; Frederick et al. Reference Frederick, Loewenstein and O'Donoghue2002). Quite another type of motivation, however, is unease at the implications of any constant discount rate for evaluation of the sufficiently far future: both the implication that the very far future matters very little indeed, and the implication that the much further future matters far less even than the ‘very’ far future. As Weitzman himself put it: ‘To think about the distant future in terms of standard discounting is to have an uneasy intuitive feeling that something is wrong, somewhere’ (Weitzman Reference Weitzman1998: 201). The result discussed in Section 9.1 therefore provides independent motivation for a claim that many had suspected for other reasons. This and other theoretical rationales for declining discount rates are surveyed in Gollier (Reference Gollier2013: ch. 7 and 8).

As is widely recognized (Strotz Reference Strotz1955; Thaler Reference Thaler1981; Loewenstein and Prelec Reference Loewenstein and Prelec1992; Rubinstein Reference Rubinstein2003), the use of a non-constant discount rate can lead to time-inconsistency in decision-making. The basic reason for this has already been mentioned (Section 7.1 above): if the ratio between the discount factors applicable to times t ₁ and t ₂ changes between two times t_A , t_B > t_A that are earlier than either t ₁ or t ₂, an agent making investment decisions at time t_B may want to reverse decisions that were optimal by the lights of the discount factors applicable at t_A . If the details of the reversal are foreseeable at t_A , game-theoretic issues arise: what should the agent at t_A do in the light of the knowledge that if he were to attempt to implement a given policy, the later agent at t_B would attempt to reverse his actions (Phelps and Pollak Reference Phelps and Pollak1968; Barro Reference Barro1999)? In particular, these issues do arise in the context of a declining discount rate that is rationalized by uncertainty $\`{a}$ la Weitzman.

9.3. Alternatives to the Expected Utility Approach

The expected-utility framework requires the decision maker to settle on a (unique) probability distribution to represent his state of uncertainty. Following Ellsberg (Reference Ellsberg1961) and Slovic and Tversky (Reference Slovic and Tversky1974), many authors argue that this is inappropriate when, as in the case of climate change, we face ‘deep uncertainty’: that is, when the available information does not even come close to singling out a single probability distribution as the uniquely rational one. The claim is that the phenomenon of ‘ambiguity aversion’ displayed in Ellsberg’s results can be (not merely actual, but) perfectly rational: that is, it can be rational to prefer known to unknown odds of significant outcomes, in a manner that is inconsistent with the conjunction of expected utility theory and any hypothesis regarding degrees of belief about the unknown-odds cases.

Non-expected utility approaches endorsing the rationality of ambiguity aversion divide into two types: those that eschew the use of probabilities altogether, and those that deal with the non-uniqueness of probabilities by some method of aggregating over the various possible (objective) probability distributions. (For useful surveys, see Al-Najjar and Weinstein Reference Al-Najjar and Weinstein2009; Etner et al. Reference Etner, Jeleva and Tallon2012; Heal and Millner Reference Heal and Millner2013.) The probability-free approaches include a straightforward maximin criterion (Wald Reference Wald1945, Reference Wald1949), a variant on maximin according to which one maximizes a weighted sum of the utilities of the worst possible and best possible outcomes (Arrow and Hurwicz Reference Arrow and Hurwicz1977), and a principle of ‘minimax regret’. Probability-based approaches include: a ‘maximin expected utility’ approach that evaluates a policy via the expected utility it has according to the prior that is least favourable to it (Gilboa and Schmeidler Reference Schmeidler1989); a ‘smooth ambiguity’ model that takes an expectation value (with respect to subjective ‘meta-probabilities’) of some increasing transform of expectation values (with respect to objective probabilities) of utility (Klibanoff et al. Reference Klibanoff, Marinacci and Mukerji2005, Reference Klibanoff, Marinacci and Mukerji2012; Epstein Reference Epstein2010); and a ‘multiplier preferences’ model (Hansen and Sargent Reference Hansen and Sargent2001, Reference Hansen and Sargent2008). The effect of ambiguity aversion on the discount rate is discussed in Gierlinger and Gollier (Reference Gollier, Koundouri and Pantelidis2008), and Traeger (Reference Traeger2012b).

11.1. Appropriateness for Matters of Global Climate Policy

The standard discounting framework, as set out in Sections 4–5, proceeds by means of first order approximation: that is, it analyses the conditions under which in the limit as the size of the ‘investment project’ under consideration goes to zero the project generates an improvement over the status quo. (This is clearest in the derivation of the Ramsey equation in the Appendix; it appears in the discussion of Sections 4–5 via the stipulation that the investment project under consideration is ‘marginal’.) As a result, the framework is entirely appropriate for analysing e.g. the small-scale policy decisions of individual government departments (who seek to take into account the climate-change implications of their policy decisions when deciding on e.g. transport policies or construction projects), but its usefulness is limited if the question under discussion is that of global climate policy – that is, how much mitigation the world as a whole ought to be undertaking. The analytical problem in the latter case is that if we are comparing two scenarios, A and B, that involve very different consumption streams, the value difference between A and B is not accurately calculated by evaluating the (marginal) rates of change of value with consumption at any given times, and multiplying by the consumption differences between A and B at the corresponding times – yet that is the closest that a naive application of the discounting framework could come to calculating that value difference. A similar point has been stressed in particular by Stern (Reference Stern2007, Reference Stern2014), who argues against overly naive application of the standard discounting framework in global-scale analyses of climate change, emphasizing that for some purposes, global-scale analyses have to return to simply evaluating the chosen value function at each of the alternative scenarios under consideration, and seeking to identify the highest-value alternative directly (as the Stern Review itself attempts to do).

It is important, however, not to overstate this point. It does not follow, from the acknowledged fact that one cannot calculate the value difference between two very different consumption paths via a ‘marginalist’ framework, that that framework is altogether irrelevant to issues of intertemporal ethics even for non-marginal, global climate policy. There are two reasons for this.

The first reason is already mentioned in Stern’s discussion: it is that, notwithstanding the sense in which choices of global climate policy ultimately require direct (non-perturbative) modelling, many of the same issues as those discussed above in the perturbative framework will also crop up, and will be amenable to much the same arguments, in any such direct modelling exercise. Any such direct analysis still has to decide on (i) the value (positive or otherwise) to assign to the rate of pure time preference in its fundamental value function, (ii) the assumptions it makes regarding future growth rates (or probability distributions over such future growth rates), and (iii) the degree of concavity in the utility function linking consumption level to individual well-being or utility; and both the ethical determinants and the implications of those decisions will still be much as discussed above.

The second reason is that the sense in which the marginalist framework is ‘inapplicable’ to the analysis of non-marginal changes is itself limited: there are, that is, some highly relevant things that one can learn even about large changes from examination of a first-order perturbative analysis (provided that the problem we face has certain structural features).

To see this, it is helpful to proceed by means of a simple, abstract analogy. Suppose that ultimately we seek the maximum of some function f; for simplicity, let f be a function of a single real-valued variable x only. Towards answering our question, let us in the first instance select one possible value ( $\overline{x}$ ) of this variable to serve as ‘status quo’. We could then start by asking a more limited question: not precisely where the maximum of f lies, but merely whether the optimum is located at a variable-value higher than, lower than or equal to $\overline{x}$ . If we know (the structural assumption in this analogy) that f is continuously differentiable, has only one stationary point and that stationary point is a maximum (rather than a minimum or a point of inflection), then a first-order perturbative analysis carried out at the ‘status quo’ point $\overline{x}$ suffices to answer this more limited question: if the derivative (i.e. the gradient) of f at $\overline{x}$ is positive (respectively, negative, zero), then the maximum must be located at a variable-value higher than (respectively, lower than, equal to) $\overline{x}$ . (The point is clearest in a graphical representation, for which see Figure 2.)

Figure 2.

The curved red line shows the graph of f as a function of x; f has a single maximum, at x ₂. We also consider randomly selected points x ₁ and x ₃ such that x ₁ < x ₂ < x ₃. The gradient of f at x ₁ (respectively, at x ₃) is positive (respectively, negative); subject to auxiliary structural assumptions, even without e.g. calculating the value of f at x ₁, x ₃ or anywhere else, or calculating the precise location of the maximum, this information on local gradients would suffice for the correct prediction that x ₁ (respectively, x ₃) is below (respectively, above) the point at which f takes its maximum value.

What this ‘first-order’ perturbative analysis cannot tell us is by how much the location of the maximum exceeds, or falls short, of x. It cannot tell us that because the maximum is the (different) variable-value at which the gradient of f is zero (i.e. the graph of f is horizontal), while the perturbative approach looks only at the gradient of f at the particular point x. This quantitative matter is certainly an important question. But an answer to the more limited question above is still somewhat informative.

Whether or not the case of climate change is analogous depends on whether or not the climate-change problem satisfies the required structural assumptions, over a sufficiently wide range of mitigation scenarios. This is a substantive question, and not one I will attempt to tackle here.Footnote ⁴ If it does, however, then relative to any ‘status quo’ consumption path – which could be the consumption path that is projected to result from any particular proposal for a course of mitigative action – a first-order perturbative analysis can tell us, given plausible background assumptions, whether the amount of mitigation in that ‘status quo’ proposal is higher than, lower than or equal to the optimum. The perturbative analysis of any given path does not tell us by how much that path falls short of, or exceeds, the optimum; but by subjecting a variety of possible consumption paths to that more qualitative question in turn, we get a good handle on the quantitative question also. In particular, via this method, the marginal approach would (given the required structural assumptions) easily be informative enough to capture most of the controversy that has raged between Stern and his critics, each of whom, as we have seen, proposes a rough specification of an optimum, and criticizes the opponent’s suggestion for involving far too much or far too little mitigation.

11.2. Distributional Issues

In Section 3, we followed the standard discounting literature in abstracting away from issues of intratemporal inequality: in place of the more fundamental value function 1, we turned to analysing instead the simpler formula 4, which latter deals only with average per capita consumption at each time. We must therefore ask whether our eventual conclusions are adversely affected by this simplification.

In the context of global climate change discussions, the answer is surely positive. As many commentators have pointed out, to the extent that we do decide to mitigate climate change, the costs of mitigation will be borne primarily by the richest of those alive today, while the benefits will accrue preferentially to the poorest of present and future people. From the point of view of ‘growth discounting’, the salient question is therefore the comparison in consumption levels between these parties. Therefore, even if the received view of consumption growth (viz., that average per capita consumption will continue to rise over the relevant time period, climate change notwithstanding) is correct, therefore, it may yet be the case that, in the relevant sense, the beneficiaries of mitigation are at best only marginally richer than those who would bear the costs. If so, clearly this would drive down the discount rate that is appropriate to evaluation of climate change mitigation.

This line of argument is strengthened further by considerations of uncertainty: combining the above line of thought with that of the Weitzman argument surveyed in Section 9.1 suggests that the relevant comparison is between the worst off possible beneficiaries of mitigation and the best off possible cost-bearers, in which case the applicable effective discount rate is sure to be negative (Fleurbaey and Zuber Reference Zuber and Asheim2012).

11.3. Changing Relative Prices

Again as we noted in Section 3, the standard framework proceeds in terms of a single real-valued index of ‘consumption’. Since ‘consumption’ in reality is of course consumption of many different goods (rice, beans, housing, clean air, access to leisure . . .), a significant amount of aggregation has already been carried out, in representing these various widely differing aspects of consumption by a single real value, rather than by a high-dimensional vector of such values.

There is nothing wrong with such aggregation in principle. However, as we flagged in Section 3, there is some danger that use of the aggregated framework might lure us into ignoring the phenomenon of changing relative prices. It is now time to explain in more concrete terms what the mistake might be, in the particular case of climate change.

The salient possibility here is that future vs present people will, in general, enjoy consumption bundles of different compositions, in such a way that while future people are (perhaps) better off overall (that is, their consumption bundle is on a higher indifference curve than the consumption bundle enjoyed by present people), the future consumption bundle involves significantly lower standards of those goods that we expect to be particularly impacted by adverse climate change (perhaps: a pleasant ambient temperature, and relative freedom from natural disasters and from certain diseases). In such a case, it might be that a given improvement to future environmental conditions generates a greater increase in overall value than would be generated by the same improvement if it occurred today – the ‘environmental discount rate’ is negative – despite the fact that future people are better off overall. In such a case, one is misled if one discounts environmental goods using the (positive) discount rate that applies to consumption.

Slightly more generally, the point is as follows. Suppose (simplifying somewhat) that we have a privileged way of representing all aspects of consumption in terms of just two indices m and e, representing, respectively, amount of ‘material’ goods consumed and amount of ‘environmental’ goods enjoyed. In terms of the indices m and e, we have a two-dimensional space of possible goods-bundles, and a utility function defined on pairs (m, e). As above, we might also define on this space a single ‘consumption’ index c. A consumption path is then a map from times to points of this space: it tells us how much consumption of each of e and m there is (and hence what level of c we are at) at each time t. Relative to any such consumption path, we can then define the shadow prices of each of e, m and c at any given time t, as the amount by which overall value (as defined by (4)) increases when an extra unit of (respectively) e, m or c becomes available at time t, while other aspects of the consumption path are held fixed. We then have a discount rate for each of the three quantities e, m, c, defined as in Section 4 from the corresponding time-profile of shadow prices. In general, all three of these discount rates will be different.

Suppose now that starting from some specified status quo consumption path, we consider a project (for example, a climate-change mitigation project) that offers an increase in environmental goods e at some future time t, in return for some decrease in material goods m at the present time 0. We wish to calculate whether this project represents an increase in overall value. One (perfectly accurate) way of analysing this matter is as follows. Define relative shadow prices between any two of e, m and c at any time t, λ _{ij, t} , as the ratio of the two shadow prices in question at that time. Then, first, convert the increase in future e to an increase in future ‘consumption’ c using the relative price of e relative to c at the future time t (λ _{ec, t} ). Second, discount back to the present time using the discount rate for consumption. Third, to compare the result to the present m-cost, we also convert that m-cost to the welfare-equivalent amount of present ‘consumption’ using the present relative prices of m and consumption (λ _{mc, 0}).

The point then is that it is easy, if one is using the aggregated ‘consumption’ framework but not thinking carefully enough, to get the conversions between e, m and c wrong, and hence to arrive at incorrect evaluations for projects of this nature. In particular, one gets them wrong if one tries to use the present relative price of environmental goods and consumption (λ _{ec, 0}) as a proxy for their future relative price λ _{ec, t} : in general, and especially in contexts of deteriorating environmental quality but rising material consumption, these two ratios will be very different (‘changing relative prices’). But this is effectively what one does if one follows the common practice of using willingness-to-pay surveys, conducted on (who else?) present people, for the purpose of placing a monetary (and thence ‘consumption’) value on future climate damages.

The case for stronger mitigation that might result from this line of thought is developed in e.g. (Guesnerie Reference Guesnerie2004; Sterner and Persson Reference Sterner and Persson2008; Gollier Reference Gollier2013: ch. 10), all of whom analyse the issue in a ‘two-good’ framework that works directly in terms of separate indices for environmental and from non-environmental (materialistic) goods throughout and corresponding separate discount rates, and eschews the use of any single, overarching index of ‘consumption’.