In Chapter 2 we argued, following distinguished economists from Frank Ramsey in the 1920s to Amartya Sen and Robert Solow more recently, the only sound ethical basis for placing less value on the utility (as opposed to consumption) of future generations was the uncertainty over whether or not the world will exist, or whether those generations will all be present. Thus we should interpret the factor e-dt in (3) as the probability that the world exists at that time. In fact this is exactly the probability of survival that would apply if the destruction of the world was the first event in a Poisson process with parameter d (i.e. the probability of an event occurring in a small time interval ?t is d?t). Of course, there are other possible stochastic processes that could be used to model this probability of survival, in which case the probability would take a different form. The probability reduces at rate d. With or without the stochastic interpretation here, d is sometimes called ‘the pure time discount rate.’ We discuss possible parameter values below.
The key concept for discounting is the marginal valuation of an extra unit of consumption at time t, or discount factor, which we denote by ?. We can normalise utility so that the value of ? at time 0 along the path under consideration is l. We are considering a project that perturbs consumption over time around this particular path. Then, following the basic criterion, equation 2, for marginal changes we have to sum the net incremental benefits accruing at each point in time, weighting those accruing at time t by ?. Thus, from the basic marginal criteria (2), in the special case (3), we accept the project if, (4) 8 ?W =???cdt > 0 0 where ? and c are each evaluated at time t, ?c is the perturbation to consumption at time t arising from the project and ? is the marginal utility of consumption where (5) ? =u'(c)e-dt If, for example, we have to invest to gain benefits then ?c will be negative for early time periods and positive later. STERN REVIEW: The Economics of Climate Change 45
? =? +d PART I: Climate Change – Our Approach
The rate of fall of the discount factor is the discount rate, which we denote by ?. These definitions and the special form of ? as in (5) are in the context of the very strong simplifications used. Under uncertainty or with many goods or with many individuals, there will be a number of relevant concepts of discount factors and discount rates.
The discount factors and rates depend on the numeraire that is chosen for the calculation. Here it is consumption and we examine how the present value of a unit of consumption changes over time. If there are many goods, households, or uses of revenue we must be explicit about choice of numeraire. There will, in principle, be different discount factors and rates associated with different choices of numeraire – see below.
Even in this very special case, there is no reason to assume the discount rate is constant. On the contrary, it will depend on the underlying pattern of consumption for the path being examined; remember that ? is essentially the discounted marginal utility of consumption along the path.
Let us simplify further and assume the very special ‘isoelastic’ function for utility c1-? 1-? u(c) = (6) ?/ c (where, for ?=1, u(c) = log c). Then
? = c-?e-dt and the discount rate ?, defined as – &? , is given by
& c (7)
(8) To work out the discount rate in this very simple formulation we must consider three things. The first is ?, which is the elasticity of the marginal utility of consumption.10 In this context it is essentially a value judgement. If, for example ?=1, then we would value an increment in consumption occurring when utility was 2c as half as valuable as if it occurred when consumption was c. The second is c/c, the growth rate of consumption along the path: this is a specification of the path itself or the scenario or forecast of the path of consumption as we look to the future. The third is d, the pure time discount rate, which generates, as discussed, a probability of existence of e-dt at time t (thus d is the rate of fall of this probability).
The advantage of (8) as an expression for the discount rate is that it is very simple and we can discuss its value in terms of the three elements above. The Treasury’s Green Book (2003) focuses on projects or programmes that have only a marginal effect relative to the overall growth path and thus uses the expression (8) for the discount rate. The disadvantage of (8) is that it depends on the very specific assumptions involved in simplifying the social welfare function into the form (3).
There is, however, one aspect of the argument that will be important for us in the analysis that follows in the Review and that is the appropriate pure time discount rate. We have argued that it should be present for a particular reason, i.e. uncertainty about existence of future generations arising from some possible shock which is exogenous to the issues and choices under examination (we used the metaphor of the meteorite).
But what then would be appropriate levels for d? That is not an easy question, but the consequences for the probability for existence of different ds can illuminate – see Table 2A.1. 10 See e.g. Stern (1977), Pearce and Ulph (1999) or HM Treasury (2003) for a discussion of some of the issues. STERN REVIEW: The Economics of Climate Change 46
PART I: Climate Change – Our Approach For d=0.1 per cent, there is an almost 10% chance of extinction by the end of a century. That itself seems high – indeed if this were true, and had been true in the past, it would be remarkable that the human race had lasted this long. Nevertheless, that is the case we shall focus on later in the Review, arguing that there is a weak case for still higher levels.11 Using d=1.5 per cent, for example, i.e. 0.015, the probability of the human race being extinct by the end of a century would be as high as 78%, indeed there would be a probability of extinction in the next decade of 14%. That seems implausibly, indeed unacceptably high as a description of the chances of extinction.
However, we should examine other interpretations of ‘extinction’. We have expressed survival or extinction of the human race as either one or the other and have used the metaphor of the devastating meteorite. There are also possibilities of partial extinction by some exogenous or man-made force that has little to do with climate change. Nuclear war would be one possibility or a devastating outbreak of some disease that ‘took out’ a significant fraction of the world’s population.
In the context of project uncertainty, rather different issues arise. Individual projects can and do collapse for various reasons and in modelling this type of process we might indeed consider values of d rather higher than shown in this table. This type of issue is relevant for the assessment of public sector projects, see, for example, HM Treasury (2003), the Green Book.
A different perspective on the pure time preference rate comes from Arrow (1995). He argues that one problem with the absence of pure time discounting is that it gives an implausibly high optimum saving rate using the utility functions as described above, in a particular model where output is proportional to capital. If d=0 then one can show that the optimum savings rate in such a model12 is 1/?; for ? between 1 and 1.5 this looks very high. From a discussion of ‘plausible’ saving rates he suggests a d of 1%. The problem with Arrow’s argument is, first, that there are other aspects influencing optimum saving in possible models that could lower the optimum saving rate, and second, that his way of ‘solving’ the ‘over-saving’ complication is very ad hoc. Thus the argument is not convincing.
Arrow does in his article draw the very important distinction between the ‘prescriptive’ and the ‘descriptive’ approach to judgements of how to ‘weigh the welfare’ of future generations – a distinction due to Nordhaus (see Samuelson and Nordhaus, 2005). He, like the authors described in Chapter 2 on this issue, is very clear that this should be seen as a prescriptive or ethical issue rather than one which depends on the revealed preference of individuals in allocating their own consumption and wealth (the descriptive approach). The allocation an individual makes in her own lifetime may well reflect the possibility of her death and the probability that she will survive a hundred years may indeed be very small. But this intertemporal allocation by the individual has only limited relevance for the long-run ethical question associated with climate change. 11 12 See also Hepburn (2006). This uses the optimality condition that the discount rate (as in (8)) should be equal to the marginal product of capital.
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?N u(C /N )e d PART I: Climate Change – Our Approach
There is nevertheless an interesting question here of combining short-term and long-term discounting. If a project’s costs and benefits affect only this generation then it is reasonable to argue that the revealed relative valuations across periods has strong relevance (as it does across goods). On the other hand, as we have emphasised allocation across generations and centuries is an ethical issue for which the arguments for low pure time discount rates are strong.
Further, we should emphasise that using a low d does not imply a low discount rate. From (8) . we see, e.g., that if ? were, say, 1.5, and c/c were 2.5% the discount rate would be, for d= 0, 3.75%. Growing consumption is a reason for discounting. Similarly if consumption were falling the discount rate would negative.
As the table shows the issue of pure time discounting is important. If the ethical judgement were that future generations count very little regardless of their consumption level then investments with mainly long-run pay-offs would not be favoured. In other words, if you care little about future generations you will care little about climate change. As we have argued, that is not a position which has much foundation in ethics and which many would find unacceptable.
Beyond the very simple case
We examine in summary form the key simplifying assumptions associated with the formulation giving equations (3) and (8) above, and ask how the form and time pattern of the various discount factors and discount rates might change when these assumptions are relaxed.
Case 1 Changing population
With population N at time t and total consumption of C, we may write the social welfare function to generalise (3) as (9) 8 W =? Nu(C / N)e-dtdt 0 In words, we add, over time, the utility of consumption per head times the number of people with that consumption: i.e. we simply add across people in this generation, just as in (3) we added across time; we abstract here from inequality within the generation (see below). Then the social marginal utility of an increment in total consumption at time t is again given by (5) where c is now C/N consumption per head. Thus the expression (8) for the discount rate is unchanged. We should emphasise here that expression (9) is the appropriate form for the welfare function where population is exogenous. In other words we know that there will be N people at time t. Where population is endogenous some difficult ethical issues arise – see, for example, Dasgupta (2001) and Broome (2004, 2005).
Case 2 Inequality within generations
Suppose group i has consumption Ci and population Ni. We write the utility of consumption at time t as i – t i i i (10) and integrate this over time: in the same spirit as for (9), we are adding utility across sub- groups in this generation. Then we have, replacing (5), where ci is consumption per head for group i, (11) ?i = u'(ci)e-dt STERN REVIEW: The Economics of Climate Change 48
PART I: Climate Change – Our Approach
as the discount factor for weighting increments of consumption to group i. Note that in principle the probability of extinction could vary across groups, thus making di dependent on i.
An increment in aggregate consumption can be evaluated only if we specify how it is distributed. Let us assume a unit increment is distributed across groups in proportions ai. Then (12) ? =?aiu'(ci)e-dt i For some cases ai may depend on ci, for example, if the increment were distributed just as total consumption, so that ai = Ci/C where C is total consumption. In this case, the direction of movement of the discount rate will depend on the form of the utility function. For example, in this last case, if ?=1, the discount rate would be unaffected by changing inequality.
If ai = 1/N this is essentially ‘expected utility’ for a ‘utility function’ given by u'( ). Hence the Atkinson theorem (1970) tells us that if {ci} becomes more unequal13 then ? will rise and the discount rate will fall if u' is convex (and vice versa if it is concave). The convexity of u' ( ) is essentially the condition that the third derivative of u is positive: all the isoelastic utility functions considered here satisfy this condition14.
For ai ‘tilted’ towards the bottom end of the income distribution, the rise is reinforced. Conversely, it is muted or reversed if ai is ‘tilted’ towards the top end of the income distribution. For example, where ai = 1 for the poorest subset of households, then ? will rise where rising inequality makes the poorest worse off. But where aN = 1 for the richest household, ? will fall if rising income inequality makes the richest better off. Note that in the above specification the contribution of individual i to overall social welfare depends only on the consumption of that individual. Thus we are assuming away consumption externalities such as envy.
Case 3 Uncertainty over the growth path
We cannot forecast, for a given set of policies, future growth with certainty. In this case, we have to replace the right-hand side of (5) in the expression for ? by its expectation. This then gives us an expression similar to (12), where we can now interpret ai as the probability of having consumption in period t, denoted as pi in equation (13). We would expect uncertainty to grow over time in the sense that the dispersion would increase. Under the same assumptions, i.e. convexity of u', as for the increasing inequality case, this increasing dispersion would reduce the discount rate over time. Increased uncertainty (see Rothschild and Stiglitz, 1976 and also Gollier, 2001) increases ? if u' is convex since ? is essentially expected utility with u' as the utility function. (13) ? =? piu'(ci)e-dt i Figure 2A.1 shows a simple example of how the discount factor falls as consumption increases over time, when the utility function takes the simple form given in equation (6). The chart plots the discount factor along a range of growth paths for consumption; along each path, the growth rate of consumption is constant, ranging from 0 per cent to 6 per cent per year. The value of d is taken to be 0.1 per cent and of ? 1.05. The paths with the lowest growth rates of consumption are the ones towards the top of the chart, along which the discount factor declines at the slowest rate. Figure 2A.2 shows the average discount rate over time corresponding to the discount factor given by equation (13), assuming that all the paths 13 This property can be defined via distribution functions and Lorenz curves. It is also called second-order stochastic domination or Lorenz-dominance: see e.g. Gollier (2001), Atkinson (1970) and Rothschild and Stiglitz (1970). 14 concave utility function. STERN REVIEW: The Economics of Climate Change 49
Discount rate 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94 97 1 4 7 PART I: Climate Change – Our Approach
are equally likely. This falls over time. For further discussion of declining discount rates, see Hepburn (2006).
Figure 2A.1 Paths for the discount factor
1.2
1
0.8
0.6
0.4
0.2
0
T i me
Figure 2A.2
Average discount rate
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0 Time
Further complications
The above treatment has kept things very simple and focused on a case with one consumption good and one type of consumer, and says little about markets.
Where there are many goods, and different types of household and market imperfections we have to go back to the basic marginal criterion specified in (2) and evaluate ?uh for each STERN REVIEW: The Economics of Climate Change 50
PART I: Climate Change – Our Approach
household taking into account these complications: for a discussion, see Drèze and Stern (1990). There will generally be a different discount rate for each good and for each consumer. One can, however, work in terms of a discount rate for aggregate (shadow) public revenue.
A case of particular relevance in this context would be where utility depended on both current consumption and the natural environment. Then it is highly likely that the relative ‘price’ of consumption and the environment (in terms of willingness-to-pay) will change over time. The changing price should be explicit and the discount rate used will differ according to whether consumption or the environment is numeraire (see below on Arrow (1966)).
Growing benefits in a growing economy: convergence of integrals.
We examine the special case (4) of the basic marginal criteria (2). The convergence of the integral requires ? to fall faster than the net benefits ?c are rising. Without convergence, it will appear from (4) that the project has infinite value. Suppose consumption grows at rate g and the net benefits at g. From (8) and (4) we have that for convergence we need, in the limit into the distant future, ˆ ?g +d > g (14) If, for example, g and g are the same (benefits are proportional to consumption) then for convergence we need, in the limit, d > (1-?)g (15) Where ?=1 and d>0, this will be satisfied. But for ?
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