if all cross-price derivatives over time are zero, then the rate of discount in any
given period (with respect to period N) is a straight weighted average of the
MRS and MRT speciWc to that period. Reinterpret Eq. (22.60) as:
r
j
pub
a
1 a
MRT
j
1
1 a
MRS
j
(24:4)
where:
a
qY
j
qp
j
qX
j
qq
j
Y
j
private sector production (use) of the good (factor) in period j.
P
j
the producer price in period j MRT
j
f
j
/f
N
.
X
j
consumption (supply) of the good (factor) in period j.
q
j
the consumer price in period j MRS
j
U
j
/U
N
.
Hence, if qX
j
/qq
j
0, or the government's investment comes entirely at the
expense of private production (investment), r
pub
MRT, as expected, and
vice versa if qY
j
/qp
j
0.
Equation (24.4) is the formula recommended by Harberger. It obviously
corresponds to the notion that r
pub
should reXect the opportunity cost of
extracting scarce resources from the private sector.
14
THE BRADFORD MODEL OF THE PUBLIC SECTOR RATE
OF DISCOUNT
The simplest cases from our second-best models appear to imply that the
appropriate public rate of discount is biased toward the private MRT, but
that would be a hasty conclusion. David Bradford developed a second-best
model that highlights the relationship between the public rate of discount and
the three factors discussed above: the opportunity cost of public funds, the
reinvestment of project beneWts, and the social rate of time preference. His
model led him to conclude that the consumers' MRS
priv
is probably closer to
the true public rate.
15
14
A. Harberger, ``The Opportunity Costs of Public Investment Financed by Borrowing,'' in
R. Layard, Ed., Cost±BeneWt Analysis, Penguin Education, Penguin Books, Ltd., Middlesex,
England, 1972. He also presents an alternative formula which aggregates across consumers and
Wrms who face diVerent MRS's and MRT's, respectively. Finally, it should be noted that
Harberger equates MRS
soc
with MRS
priv
. He does not believe there is a signiWcant divergence
between the two.
15
D. Bradford, ``Constraints on Government Investment Opportunities and the Choice
of the Discount Rate,'' American Economic Review, December 1975. Bradford's principal
24. THE RATE OF DISCOUNT FOR PUBLIC INVESTMENTS 743
Bradford's model is more general than the ones we have been using
because it permits any underlying market structure. Furthermore, his
model does not necessarily require the existence of distorting taxation, nor
even that the government's budget constraint must balance, assumptions that
we used repeatedly in the theoretical chapters on second-best expenditure
theory. Rather, his analysis is second-best simply because the government has
certain investment opportunities that are not open to the private sector. In
terms of our previous models, if the government produces some output Z
i
,
there cannot be a corresponding private sector output Y
i
. Of course, our
previous models were general enough to consider this possibility, but they
placed a considerable number of restrictions on the underlying policy envir-
onment. Bradford avoids any speciWc restrictions by deWning the relationship
between private investment and the future stream of consumption broadly
enough to encompass any speciWc set of market or policy assumptions one
might choose to make.
Following Bradford, let V
t
the present value of the stream of con-
sumption, discounted at the social MRS, that is generated by $1 of private
investment at time t. The stream of consumption beneWts begins in time t 1.
Presumably, V
t
could be calculated for any speciWc model, such as those in
Parts II and III of this text. The calculations may be extremely complex, but
V
t
would nonetheless be well deWned for any given intertemporal general
equilibrium model.
Bradford completes his model by assuming that:
1. The one-period rate of return on private investment, the MRT, is
constant over time at rate r. $1 of private investment at time t yields $(1 r)
of income in time t 1.
2. The government's objective function is the discounted stream of
aggregate consumption over time, where the discount rate is the social
MRS, equal to i. Assume that the government's MRS
soc
equals the consu-
mer's private MRS
priv
. SpeciWcally, Bradford assumes that the objective
function is additively separable over time with constant undiscounted mar-
ginal utilities. That is,
UC
1
, ...,C
N
P
N
j1
1 i
j
f
j
C
j
(24:5)
with
qf
j
qC
j
kj 1, ...,N
contribution is that his model includes the reinvestment of project beneWts. In other respects,
his approach was already well represented in the literature (e.g., S. Marglin, ``The Opportunity
Costs of Public Investment''; M. Feldstein, ``The Inadequacy of Weighted Discount Rates'').
744 THE BRADFORD MODEL OF THE PUBLIC SECTOR RATE OF DISCOUNT
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