1 Introduction
1.1 The discounting issue: weighting future generations
1.2 Fundamental utilitarianism
1.3 Extinction discounting
1.4 Outline
2 Discounting what?
2.1 The social rate of discount for future consumption
Thereafter Sect. 4 of Arrow (1999a) starts as follows:\( \rho \) is the of pure time preference (if any), \( \theta \) is the elasticity of marginal utility with respect to income, and g is the of growth of consumption per capita ...\( \rho = 0\) implies equal treatment of present and future.
In [the] formula [\(r = \rho + \theta g\)], the second term, \( \theta g\), is, I think, fairly uncontroversial. If future individuals are going to be better off than we are, then our willingness to sacrifice on their behalf is certainly reduced. It would require a greater rate of return to justify our depriving ourselves of consumption.
In the ensuing discussion of the term \( \theta g\), Arrow (1999a) considers how to value an increment in a good in the future, relative to an increment now. This is the discount factor, which we denote by \( \beta \), for that good at that time. In the continuous time model that Arrow uses, the proportionate rate of decrease of the discount factor is given by \( - {{\dot{\beta }}} / \beta = - \frac{{{\,\mathrm{d \!}\,}}}{ {{\,\mathrm{d \!}\,}}t} \ln \beta \). It is the discount rate for that good at that time; it clearly depends on both the good and the time. In our view, the focus in economic assessments should be on the discount factor, as that is the key shadow price, relative to now, which is needed to find the marginal present value of any change in costs or benefits occurring at any specific time in the future. When we need to evaluate a stream of costs and benefits over time, we can consider the net present value (NPV) of the whole stream, with the costs and benefits at each time t weighted by the discount factor \( \beta (t)\). That is, at time 0 one considers \( \text{ NPV } := \int _0^T \beta (t) \, b(t) {{\,\mathrm{d \!}\,}}t\), where b(t) denotes net benefit at time t, and T denotes the terminal time. When allowing for the inevitable uncertainty surrounding future costs and benefits, one approach is to consider their expected discounted value.But the presence of pure time preference, denoted by \( \rho \), has been very controversial. The English economists, in particular, have tended to be very scornful of pure time preference.
2.2 Discounting the welfare of future generations
In his celebrated paper on optimal saving, Ramsey (1928, p. 261) famously started out by following the spirit of Sidgwick when he refused to discount the welfare of future generations:“How far we are to consider the interests of posterity when they seem to conflict with those of existing human beings? It seems ...clear that the time at which a man exists cannot affect the value of his happiness from a universal point of view; and that the interests of posterity must concern a Utilitarian as much as those of his contemporaries, except in so far as the effect of his actions on posterity—and even the existence of human beings to be affected—must necessarily be more uncertain.”
Arrow (1999b) not only quotes this passage from Ramsey, but also adds two later quotes from other English economists. The first is from Pigou (1932, p. 25) stating that pure time preference “implies ...our telescopic faculty is defective.” The second is Harrod’s (1948, p. 40) claim that “[P]ure time preference [is] a polite expression for rapacity and the conquest of reason by passion.”9One point should perhaps be emphasised more particularly; it is assumed that we do not discount later enjoyments in comparison with earlier ones, a practice which is ethically indefensible and arises merely from the weakness of the imagination.8
2.3 Discounting our own future
2.4 The strong argument
Why then not embrace the idea of zero time perspective? Koopmans in several classic papers (1960, 1964) gave a crushing answer; see also Brown and Lewis (1981) for a more general treatment. The argument seems recondite. Koopmans considers a world which lasts forever. Therefore choice (including ethically-based choice) is based on a preference ordering over infinite-dimensional consumption streams. He argues that if the ordering is continuous and also sensitive (i.e., if one stream is never worse than another and is better at one or more time points, then it must be strictly preferred), it must display impatience.
Thus, Arrow concludes that without impatience the optimal saving rate could become arbitrarily close to 100%. A similar conclusion emerges from the cake-eating example described by Gale (1967, p. 4, Example 2).A simple restatement of his reasoning can bring out the essential point. I confine myself to the intertemporally separable case. Imagine initially that output consists of a constant stream of completely perishable goods. There can be no investment by definition. Now imagine that an investment opportunity occurs, available only to the first generation. For each unit sacrificed by them, a perpetual stream of \( \alpha \) per unit time is generated. If there were no time preference, ...we can say that given any investment, short of the entire income, a still greater investment would be preferred.
Thus, it seems that Koopmans (and Diamond) started out by considering only consumers who discount their own future selves, as discussed in Sect. 2.3.11 Nevertheless, it was natural for Arrow to consider the obvious extension to social choice theory, with an infinite series of successive generations. Indeed, this follows the tradition of the later works by Koopmans (1965, 1967), who even devotes part of these surveys to the Ramsey case when the welfare of future generations remains undiscounted. Then the same mathematical analysis which, under some conditions, shows that a consumer cannot treat equally consumption in an infinite number of periods, also rules out intergenerational equity, in the sense of treating all future generations equally.This study started out as an attempt to formulate postulates permitting a sharp definition of impatience, the short term Irving Fisher has introduced for preference for advanced timing of satisfaction. To avoid complications connected with the advancing age and finite life span of the individual consumer, these postulates were set up for a (continuous) utility function of a consumption program extending over an infinite future period. The surprising result was that only a slight strengthening of the continuity postulate ...permits one to conclude from the existence of a utility function satisfying the postulates, that impatience prevails at least in certain areas of the program space.
2.5 Two kinds of domain restriction
2.6 The weak argument
A very similar argument is adduced in Arrow (1999b), where he adds:I therefore conclude that the strong ethical requirement that all generations be treated alike, itself reasonable, contradicts a very strong intuition that it is not morally acceptable to demand excessively high savings rates of any one generation, or even of every generation.
And in Arrow (2007, p. 4) he writes:Not merely is saving arbitrarily close to 100% unacceptable but very high sacrifices are also. I call this the weak Koopmans argument.
This kind of argument, and its relation to intergenerational equity, were discussed by Asheim et al. (2001) and then Asheim and Buchholz (2003). Their main result demonstrates that, given any efficient and non-decreasing consumption allocation, there exists a utility function for which that allocation is a unique maximum of the undiscounted sum of utilities over all future generations. This relies, however, on a key “technological” assumption that any efficient and non-decreasing consumption allocation maximizes the finite present discounted value of consumption with discount factors that decrease over time. This assumption seems to rule out the kind of climate emergency that the world may be facing currently.Tjalling Koopmans pointed out in effect that the savings rates implied by zero time preference are very much higher than those we observe. (I am myself convinced by this argument.)
Thus, KA’s reasoning included the recognition that, even with substantial pure-time discounting, unmanaged climate change has the potential to cause damage severe enough to justify strong action.14Many have complained about the Stern Review adopting a value of zero for \( \rho \), the social rate of time preference. However, I find that the case for intervention to keep CO\(_2\) levels within bounds (say, aiming to stabilize them at about 550 ppm) is sufficiently strong as to be insensitive to the arguments about \( \rho \).13
3 An impartial spectator’s fundamental utility
3.1 A universal domain of personal consequences
3.2 Fundamental cardinal utility
3.3 General discrete lotteries and bounded utility
3.4 A unique normalized fundamental utility function
4 Utilitarianism for an impartial benefactor
4.1 Social consequences as personal consequence profiles
4.2 Biased and extended original positions
4.3 Expected utility from a biased original position
4.4 An original position with suppes equity
5 Extinction discounting of future generations
5.1 Generational structures
5.2 A hazard process of extinction and survival
5.3 Extinction adjusted intergenerational equity
5.4 Utilitarianism with an infinite population
5.5 Extinction discounted utilitarianism
6 Sustainable preferences
6.1 Sustainability
Decades before, Hicks (1946, p. 174) had a similar idea when he defined an individual’s “income” as...development that meets the needs of the present without compromising the ability of future generations to meet their own needs.
In this spirit, and following Solow (1991, 2012), sustainability might be defined as giving each generation access to an opportunity set that allows it to be no worse off than it would have been with the opportunity set that was available to any of its predecessors. This suggests trying to maximize the initial generation’s welfare level subject to monotone sustainability—i.e., requiring successive generations’ welfare levels to be non-decreasing over time.17...the maximum amount of money which the individual can spend this week, and still expect to be able to spend the same amount in real terms in each ensuing week.