This paper shows that the emphasis on a social time preference rate (defined as the sum of a pure time preference rate and the product of the elasticity of marginal valuation and the growth rate) in social evaluations where money values are discounted using the social time preference rate, is not advisable. It can give an entirely different, and arbitrary, ranking of alternative streams compared with the direct use of the pure time preference rate to discount ‘social welfare’ in each period (where social welfare is a — usually isoelastic — function of money values).

John Creedy

English

Research Papers in Economics

  ISSN  0819-2642 
  ISBN 0 7340 2647 1 
 
 
 
 
 
 
THE UNIVERSITY OF MELBOURNE 
 
DEPARTMENT OF ECONOMICS 
 
 
RESEARCH PAPER NUMBER 989 
 
 
MARCH 2007 
 
 
Discounting and the Social 
Time Preference Rate 
 
by 
 
John Creedy 
 
   
Department of Economics 
The University of Melbourne  
Melbourne      Victoria     3010 
Australia. 
 Discounting and the Social Time Preference
Rate∗
John Creedy
The University of Melbourne
Abstract
This paper shows that the emphasis on a social time preference rate
(deﬁned as the sum of a pure time preference rate and the product
of the elasticity of marginal valuation and the growth rate) in social
evaluations where money values are discounted using the social time
preference rate, is not advisable. It can give an entirely diﬀerent, and
arbitrary, ranking of alternative streams compared with the direct use
of the pure time preference rate to discount ‘social welfare’ in each
period (where social welfare is a — usually isoelastic — function of
money values).
∗Ih a v eb e n e ﬁted from discussions with Ross Guest and Grant Scobie, and comments
from Robert Dixon, Tim Helm and Guyonne Kalb.
11 Introduction
The aim of this paper is to draw attention to a problem associated with the
use of the social time preference rate, which seems to have been overlooked in
the literature on cost-beneﬁt analysis and project appraisal. The social time
preference rate, δ say, is a central concept in cost-beneﬁta n a l y s i si n v o l v i n g
the evaluation of alternative time proﬁles of, say, consumption. It is deﬁned
a st h es u mo ft h ep u r et i m ep r e f e r e n c er a t e ,ρ, and the product of the elas-
ticity of marginal valuation, ε, and the growth rate of consumption, g: hence
δ = ρ+εg. It is therefore often stressed that an extreme assumption of zero
pure time preference does not imply zero discounting, in view of the pres-
ence of the term εg.1 The values of ρ and ε reﬂe c tt h ev a l u ej u d g e m e n t so f
an independent judge or policy maker.2 Hence, cost-beneﬁt analyses should
in principle include a sensitivity analysis, involving the evaluation of time
proﬁles for a range of values of ρ and ε.
The social time preference rate has in fact been the focus of considerable
attention recently, partly because of its use by Stern (2006) in examining the
economic eﬀects of climate change and abatement policies.3 Much attention
has been given to the choice of values of ρ and ε.4 Several of the critics
have stressed the point made above, that the appropriate approach is not,
as in Stern (2006), to impose particular values but to carry out sensitivity
analyses, so that readers, with their own possibly diﬀerent value judgements,
can make up their own minds. However, the focus of the present paper is on
t h eu s eo ft h es o c i a lt i m ep r e f e r e n c er a t ei t s e l f .
The social welfare function, summarising the value judgements of the
1An arbitrary list of texts on cost-beneﬁt analysis which discuss the social time pref-
erence rate includes Brent (1990, pp. 71-72, 92; 2003, pp. 166-168), Hanley and Spash
(1993, pp. 128-130), Bateman et al. (2002, pp. 55-58), Layard and Glaister (1994, pp.
33-35), Dasgupta and Pearce (1972, pp. 141-143), Pearce and Ulph (1998) and Lind (1982,
p. 89).
2Despite the inclusion of the word ‘social’, the social time preference rate is not an
attribute of a society, in view of the well-known problem of aggregating preferences.
3See also the background paper by Hepburn (2006). Examples of criticisms of the
discount rate used by Stern include Nordhaus (2006), Dasgupta (2006) and Carter et al.
(2006).
4However, a discussion concentrating on gt is by Weitzman (2007).
2decision maker and underlying evaluations based on the social time preference
rate, takes the form
PT
t=1 U (ct)
³
1
1+ρ
´t−1
,w h e r ect represents consumption
in period t =1 ,...,T and U (ct) is the weight attached to ct by the judge; the
more concave is U, the greater the judge’s aversion to variability over time (as
distinct from time preference). This is discussed further in section 2, where it
is shown that the concept of the time preference rate arises from consideration
of the diﬀerent question of determining the optimal time stream of ct, subject
to a wealth constraint. A comparison between the direct use of the social
w e l f a r ef u n c t i o na n dt h eu s eo ft h es o c i a lt i m ep r e f e r e n c er a t ei sp r o v i d e di n
section 3. It is argued that evaluations should be based directly on the form
of welfare function mentioned above, rather than simply discounting values
of ct using the discount rate ρ+εg. The potentially misleading nature of the
latter approach is illustrated using a numerical example in section 4.
2 A Social Welfare Function
Suppose it is required to evaluate a time stream c1,...cT of consumption. For
convenience ct can be considered as the aggregate consumption in a society
with constant population size and composition. Such an evaluation cannot
avoid the use of value judgements, and the usual approach is to examine
the implications of adopting a range of value judgements, using an additive
Paretian social welfare function — representing the views of an independent
j u d g e—w h i c ht a k e st h ef o r m :
W =
T X
t=1
U (ct)
µ
1
1+ρ
¶t−1
(1)
where U (ct) is the weight attached to period t’s consumption by the judge,
and ρ is the rate of pure time preference. The weighting function U is often
called a utility function — although this terminology is misleading — and hence
the time preference rate is sometimes also called a ‘utility discount rate’.
Consideration of alternative value judgements regarding U is facilitated
by the use of the iso-elastic form:
U (ct)=
c
1−ε
t
1 − ε
(2)
3The term ε 6=1measures the degree of constant relative aversion to vari-
ability on the part of the judge. Those who refer to U as a utility function
typically refer to ε as the constant (absolute value of the) elasticity of mar-
ginal utility. Hence alternative value judgements — within the context of this
class of welfare functions — can be examined by investigating W for a range
of values of ε and ρ.
However, this is not always the way analyses proceed. Suppose that
instead of considering an exogenous time stream of ct, a ‘social planner’ has
to determine the optimal time path by maximising a social welfare function
of the form in (1). This is maximised subject to a budget constraint, which
can be regarded as taking the simple form:
T X
t=1
µ
1
1+r
¶t−1
ct = Y (3)
where Y represents a measure of the present value of resources available for
consumption over the period, and r is the rate of interest in a ‘perfect’ capital
market. By forming the Lagrangean for this problem it can be shown that
the ﬁrst order condition for period t can be expressed as:
gt =
1
ε
(r − ρ) (4)
where gt is the optimal proportional growth rate of consumption at t.T h i se x -
pression is known as the Euler equation for optimal consumption: it describes
the time path of consumption for which (1) is maximised. This equation plays
a prominent role in optimal growth models, where r can be regarded as being
determined by, for example, the marginal product of capital — depending on
the precise nature of the model considered.
Having obtained the result in (4) for optimal consumption, the standard
approach is to move swiftly to the diﬀerent context of cost-beneﬁte v a l u a -
tions, along the following lines. Another way of expressing the Euler equation
is to rearrange (4) to give r = ρ + εgt. In the context of social evaluations
of given time streams, it is this rearrangement of (4) that plays a substantial
role. Considerable attention is given to the right hand side, which is called
4the social time preference rate, δ, and is thus deﬁned as:
δ = ρ + εgt (5)
T h es o c i a lt i m ep r e f e r e n c er a t ec a nt h e r e f o r ev a r yo v e rt i m e ,d e p e n d i n go n
the behaviour of gt. In practice, it is often assumed that g, the growth rate of
aggregate consumption, is constant. The latter is typically taken as the long
run or average rate of growth over the relevant period. The value of δ is the
discount rate used to evaluate the present value of the time stream of ct,f o r
t =1 ,...,T, rather than the time stream of U (ct).H e n c eδ is often called the
‘consumption discount rate’. In the context of cost-beneﬁt analyses where
money values of an exogenous consumption stream are evaluated, then the
social time preference rate, δ, does not need to be set equal to the market
rate of interest, so that δ d o e sn o th a v et oe q u a lr. This in turn means that
no degree of freedom is lost in the choice of parameters ρ and ε (for a given
g). These two terms reﬂect the value judgements of the independent judge
whose preferences are summarised by (1).
The social time preference rate is sometimes called the ‘consumption dis-
count rate’ because it is applied to money values of consumption in each
period, whereas the pure time preference rate is sometimes called the ‘utility
discount rate’ because it is applied to weighted consumption values, with
the weighting function described as a ‘utility function’. In the literature, it
is simply taken for granted that discounting money values according to the
rate in (5) is appropriate, rather than starting from the more fundamental
social welfare function. The following section therefore compares the two
approaches.
3 Alternative Evaluation Methods
The standard approach in cost-beneﬁt evaluations, discussed in the previous
section, is to use a social time preference rate, as in (5), to discount money
ﬂows ct over a speciﬁed period. This produces a ‘social evaluation’ using W∗,
where:
W
∗ =
T X
t=1
ct
µ
1
1+δ
¶t−1
(6)
5with δ = ρ + εg. It is taken for granted that this function gives the same
ranking of projects as does the social welfare function in (1). This section
compares the two evaluation methods explicitly.
In comparing the two forms of evaluation, it is convenient to begin with
the most favourable case, that is where consumption does in fact grow at the
constant proportional rate, g. Hence ct = c1 (1 + g)
t−1 , for t =1 ,...,T,a n d
substitution gives:
W =
T X
t=1
©
c1 (1 + g)
t−1ª1−ε
1 − ε
µ
1
1+ρ
¶t−1
(7)
Rearrangement of this expression gives:
W =
T X
t=1
c
1−ε
1 (1 + g)
(t−1)−ε(t−1)
1 − ε
µ
1
1+ρ
¶t−1
(8)
and:
W =
c
−ε
1
1 − ε
T X
t=1
c1 (1 + g)
(t−1)
µ
(1 + g)
−ε
1+ρ
¶t−1
(9)
Furthermore, using the approximation (1 + ρ)(1+g)
ε =1+ρ + εg,t h i s
becomes:
W =
c
−ε
1
1 − ε
T X
t=1
ct
µ
1
1+ρ + εg
¶t−1
(10)
and:
W =
c
−ε
1
1 − ε
W
∗ (11)
This ﬁnal results demonstrates that it is not in fact correct to believe that
W∗, obtained by discounting money values of consumption at the social time
preference rate, coincides with W, obtained by discounting U (ct) at the pure
time preference rate ρ.
For given ε, W∗ automatically gives the same ranking as W only if ε<1
a n dt w oc o n s u m p t i o ns t r e a m s ,w i t hd i ﬀerent growth rates, have the same
initial value of consumption. Otherwise, inconsistencies can arise.
For example, suppose ε =2 , and two consumption streams A and B give
values of WA = −50 and WB = −100. Hence using this criterion, stream
A is judged to be superior to B. If cA,1 =1 0 , equation (11) shows that
6W∗
A = −100WA =5 0 0 0 . if cB,1 =2 , W∗
B = −4WB =4 0 0 ,a n dt h er a n k i n gb y
W∗ agrees in this case. However, if the initial consumption values of the two
streams are both equal to 2, then the values of W∗
A and W∗
B are 200 and 400
respectively, and the ranking is reversed.
Suppose instead that ε =0 .5, and WA =8 0while WB =5 0 ,s ot h a t
stream A is again judged to be superior to stream B. However, if cA,1 =4
and cB,1 =1 6 , it can be seen that WA =
¡
2/
√
4
¢
W∗
A and W∗
A is also equal to
80,b u tW∗
B =5 0 /(2/
√
16) = 100: hence the ranking is reversed when W∗ is
used.
One way to view the comparisons is to recognise that if the ‘utility’ func-
tion is instead U (ct)=kc
1−ε
t , with the constant k = cε
1, then the values of
W and W∗ are equal for any given consumption stream. This may seem like
an innocent monotonic transformation of U. However, the social evaluation
functions are essentially cardinal: they are not invariant to monotonic trans-
formations of U, since they are expressed explicitly as additive functions of
diﬀerent U values. Furthermore, the function kc
1−ε
t gives negative marginal
utility if ε>1, and so must be ruled out.
Furthermore, in practice it is likely that g is not constant over the rele-
vant period. The assumption of constant g introduces a further ‘error’ when
discounting money values using ρ + εg, compared with discounting U with
ρ. Allowance for changing growth rates is automatic in the latter case.
4 A Numerical Example
M o r ec o m p l e xc o m p a r i s o n sm a yr e s u l tf r o mm o r ev a r i a b l et i m ep r o ﬁles, mak-
ing the choice of alternative streams more sensitive to the choices of ε and ρ.
Consider Figure 1, where time stream A results from a constant growth rate
of 2.3 per cent (staring from 10 units), but proﬁle B results from a ﬁxed trend
rate of growth (of 1.8 per cent, starting from 4 units) combined with a cycli-
cal growth component having an amplitude of 5 per cent and a wavelength
of 165 periods. From the multiple intersections, it is likely that stream B has
t h eh i g h e s tv a l u eo fW (C) f o rb o t hl o wa n dh i g hv a l u e so fρ,w h i l es t r e a mA
is likely to dominate for intermediate values, though the precise values again
7again likely to be sensitive to the choice of ε.A ne x a m p l ei sg i v e ni nF i g u r e
2, for a value of the elasticity of marginal valuation, ε,o f0 . 6 .
A
B
C
o
n
s
u
m
p
t
i
o
n
Time
A
B
C
o
n
s
u
m
p
t
i
o
n
Time
Figure 1: Alternative Time Proﬁles
Evaluations of the two time proﬁles using W∗ are unlikely to give the
same ranking. For example, Figure 3 shows the present value of the time
streams of consumption shown in Figure 1, for ε =0 .6,u s i n gW∗,t h a ti s
with money values discounted using the rate ρ + εg and with g set equal to
the trend rate of growth. It can be seen that proﬁle A dominates proﬁle B
for all values of ρ whereas, using the same value of ε =0 .6, comparisons of W
depend signiﬁcantly on the value of ρ used, as illustrated in Figure 2 above.
5 Conclusions
This paper has shown that the emphasis on a social time preference rate,
expressed in terms of ρ + εg, in social evaluations where money values are
discounted using the social time preference rate, is not advisable. It can give
an entirely diﬀerent, and arbitrary, ranking of alternative streams compared
with the discounting of U using ρ.
80.01            0.02           0.03            0.04      
P
r
e
s
e
n
t
 
V
a
l
u
e
0.01            0.02           0.03            0.04      
P
r
e
s
e
n
t
 
V
a
l
u
e
A: epsilon = 0.6
B: epsilon = 0.6
Rho
Figure 2: Rankings for Epsilon of 0.6
A: epsilon = 0.6
B: epsilon = 0.6
P
r
e
s
e
n
t
 
V
a
l
u
e
g ρε +
A: epsilon = 0.6
B: epsilon = 0.6
P
r
e
s
e
n
t
 
V
a
l
u
e
g ρε +
F i g u r e3 :C o m p a r i s o n sU s i n gt h eS o c i a lT i m eP r e f e r e n c eR a t e
9References
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11

Discounting and the Social Time Preference Rate

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