Eq. (22.88) applies to the rate of discount, as it does to all government
shadow prices. Hence, intratemporal social welfare rankings may possibly
overwhelm the pure eYciency aspects of the public discount rate, which are
all these models consider.
EMPIRICAL EVIDENCE ON THE PUBLIC RATE OF DISCOUNT
What Do the Experts Say?
Martin Weitzman conducted a survey of economists in which he asked them
what single real interest rate they would recommend for discounting the
beneWts and costs of environmental projects that are designed to reduce the
harmful eVects of changes in global climate. He received 2160 replies from
economists in 48 diVerent countries. The replies demonstrated a large diVer-
ence of opinion among economists about the appropriate rate of discount.
The vast majority of replies were from 1 to 6%, but the overall range was 3
to 27%. The mode was 2%, the mean just under 4%, and the standard
deviation just under 3%.
29
Weitzman observed that the distribution of responses was reasonably
well approximated by the gamma distribution truncated at zero (only three
responses were negative). This led him to propose a declining discount rate
over time by the following argument.
Let:
Z(t) net project beneWts at time t
A(t) the discount factor at time t.
Assuming a continuous time framework (to conform with the continuous
gamma distribution), the present value of the investment is
PV
1
0
AtZtdt (24:31)
With a single constant discount rate x, which all respondents were asked to
assume, the discount factor would be
Ate
xt
(24:32)
But given uncertainty over the appropriate discount rate from the econo-
mists' replies, Weitzman argues that it makes sense to think of each
29
M. Weitzman, ``Gamma Discounting,'' American Economic Review, March 2001. The
sample statistics are reported on p. 268, with the complete distribution of responses listed in Table
1, p. 268. Weitzman notes that many economists were reluctant to name a single rate, probably
because they understand that the appropriate rate is model dependent. Weitzman persisted in
asking them to select a single rate, which most of the economists were willing to do.
24. THE RATE OF DISCOUNT FOR PUBLIC INVESTMENTS 753
economist's answer as a single draw, x
j
, from the probability density function
of rates f(x). According to this view, the appropriate discount factor is
A(t)
1
0
e
xt
fxdx (24:33)
The discount factor is a weighted-average function of the various rates
proposed by the economists. The weights are the probabilities that each
rate is the correct rate, with the probabilities given by the density function
of the proposed rates, in this case the gamma distribution. The gamma
distribution has the form:
f(x)
b
a
Ga
x
a1
e
bx
(24:34)
Thus,
A(t)
b
a
Ga
1
0
x
a1
e
btx
dx (24:35)
The value of the integral is such that A(t) has the simple form:
A(t)
b
b t
a
(24:36)
For the gamma distribution,
m
a
b
(24:37)
s
2
a
b
2
(24:38)
Therefore,
a
m
2
s
2
(24:39)
b
m
s
2
(24:40)
and
A(t)
1
1 ts
2
.
m
m
2
s
2
(24:41)
The instantaneous Xow rate of discount R(t) is
754 EMPIRICAL EVIDENCE ON THE PUBLIC RATE OF DISCOUNT
Rt
_
At
At
(24:42)
which from Eq. (24.41) is
Rt
m
1 ts
2
.
m
(24:43)
Equation (24.43) implies that the appropriate instantaneous rate of discount
should decline over time if there is any uncertainty surrounding the appro-
priate rate. This is true even if all economists believe the rate should be
constant over time, as they were asked to assume.
30
In particular, the instant-
aneous rate begins at rate m at time zero and then declines toward zero as
t !1. For the sample of responses in Weitzman's survey, the (approximate)
discount rates for future time periods turn out to be
31
1±5 years 4%
6±25 years 3%
26±75 years 2%
76±300 years 1%
> 300 years 0%
The intuition behind the declining rates can be seen from Eq. (24.33). As
time increases, the present value of the weighted higher rates necessarily
decreases in importance relative to the weighted lower rates.
Discounting Within the Federal Government
The ``experts' '' uncertainty about the appropriate public rate of discount is
apparently shared by federal policymakers. Bazelon and Smetters recently
published an overview of discounting practices within the federal govern-
ment. They characterized the range of discount rates in use as ``striking.''
32
One would hope that a common discount rate (or set of declining rates ala
Weitzman), would be used to evaluate all government investment projects,
but this turns out to be not even approximately true.
The OYce of Management and Budget (OMB) appears to inject some
uniformity in the area of cost±beneWt analysis. Its regulation OMB-94 in-
structs all federal agencies to undertake a cost±beneWt analysis using a 7%
real discount rate for its investments and regulations. The 7% rate was chosen
30
Notice that R(t) m for all time periods if s
2
0, that is, if all economists were to pick
the same rate m .
31
M. Weitzman, ``Gamma Discounting,'' American Economic Review, March 2001, Table
2, p. 270.
32
C. Bazelon and K. Smetters, ``Discounting Inside the Washington, D.C. Beltway,''
Journal of Economic Perspectives, Fall 1999. The characterization is on p. 219.
24. THE RATE OF DISCOUNT FOR PUBLIC INVESTMENTS 755
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