from assuming e/b 1 are within a range of 11% (Bradford: 5%). Errors of
this magnitude are probably within the range of cost estimation errors, as
Bradford contends.
OTHER VIEWS ON THE APPROPRIATE RATE OF DISCOUNT
Bradford's conclusion is hardly the last word on the appropriate rate of
discount. Indeed, we have already presented a number of theoretical cases
in which the appropriate discount factor is either the MRT or a weighted
average of the MRS and MRT. Furthermore, Bradford's V
t
calculation
assumes a simple model in which there is essentially only one good and a
constant savings rate out of private sector income. Preferences are also
severely restricted in other ways. More importantly, he assumes in calculating
V that private investment yields returns r for one period only, and nothing
thereafter. Hence, the maximum value of V, which occurs if all returns are
consumed (s 0),is(1 r)/(1 i), a number still reasonably close to one.
Many economists' using his basic framework (e.g., Marglin, Feldstein, Sjaas-
tad, and Wisecarver
18
) assume private investment is a perpetuity yielding r
forever, which implies a V r/i with zero savings. This V is almost certainly
much greater than 1, perhaps in the 2.5±4.0 range. Hence, under the perpetu-
ity assumption, one would certainly not be willing to ignore the scaling factor
e/b. The ``best'' assumption for private investment undoubtedly lies some-
where between the extremes of one-period and perpetual yields. In any event,
results such as the Diamond±Mirrlees theorem suggest that the limits implied
for the scaling factor in Bradford's model may be wide of the mark for many
second-best environments, especially those with variable savings rates. This
merely reemphasizes the general proposition demonstrated throughout the
text that second-best results can be extremely sensitive to underlying assump-
tions.
Despite these reservations, Bradford's model is comprehensive enough to
serve as a convenient vehicle for presenting some of the conXicting views in
the literature.
Marglin±Feldstein: The Social Rate of Time Preference
The Marglin±Feldstein view that the appropriate present value calculation
consists of discounting project beneWts at the social rate of time preference
and adjusting project costs by a shadow price reXecting second-best distor-
18
S. Marglin, ``The Opportunity Costs of Public Investment,'' Quarterly Journal of Eco-
nomics, May 1963; M. Feldstein, ``The Inadequacy of Weighted Discount Rates,'' in R. Layard,
Ed., Cost±BeneWt Analysis, Penguin Education, Penguin Books, Ltd., Middlesex, England, 1972;
L. Sjaastad and D. Wisecarver, ``The Social Cost of Public Finance,'' Journal of Political
Economy, June 1977.
748 OTHER VIEWS ON THE APPROPRIATE RATE OF DISCOUNT
tions follows directly from Eq. (24.11). To see this, suppose all investment
costs occur immediately and yield a stream of beneWts forever. Under these
assumptions, e
t
0, t 1, and Eq. (24.11) becomes;
PV
P
N
t1
d
t
b
t
e
b
E
0
> 0 (24:16)
This is essentially the original Marglin result,
19
with the shadow price of
project costs equal to Bradford's scale factor, e/b. The only diVerence is that
Marglin is not so willing to ignore the shadow price, given his assumption
that private investment is a perpetuity. For instance, ignoring reinvestment
(a 0), and assuming all private beneWts are consumed,
e
b
e 1 aaV 1 aa
r
i
(24:17)
which may diVer substantially from 1.
Feldstein extended Marglin's analysis to include cash deWcits whenever
they occur.
20
His recommended procedure also follows immediately from
Eq. (24.11), although care must be taken to distinguish between true project
costs and out-of-pocket costs. Suppose that some project beneWts are sold
and some costs are paid for by the government each period.
Let:
b
t
true project beneWts in time t.
E
t
true project costs in time t.
R
t
project revenues in time t.
C
t
project cash payments in time t.
Net consumption beneWts are (b
t
R
t
), and net transfers to consumers are
(C
t
E
t
). The cash deWcit is (C
t
R
t
), to which the shadow price of funds
must be applied. Assuming no reinvestment of net project beneWts (a 0),
Feldstein's version of Bradford's formula is
PV
P
1
t0
d
t
h
b
t
R
t
C
t
E
t

1 aaV
t
C
t
R
t

i
> 0
(24:18)
Rearranging terms:
PV
P
1
t0
d
t
h
b
t
E
t
V 1aC
t
R
t

i
> 0 (24:19)
19
S. Marglin, ``The Opportunity Costs of Public Investment,'' Quarterly Journal of Eco-
nomics, May 1963; Marglin uses a continuous time model.
20
M. Feldstein, ``The Inadequacy of Weighted Discount Rates,'' in R. Layard, Ed., Cost±
BeneWt Analysis, Penguin Education, Penguin Books, Ltd., Middlesex, England, 1972.
24. THE RATE OF DISCOUNT FOR PUBLIC INVESTMENTS 749

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