we simplify still further
to write
(3)
8
W =?u(c)e-dtdt
0
We assume here that there is just one individual at each point in time (or a group of identical
individuals) and that the utility or valuation function is unchanging over time. We introduce
population and its change later in the discussion.
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.
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? =? +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.
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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
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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.
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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
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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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