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ALFRED
P.
WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
SOCIAL VT^UATIOSI OF PR3JECTS:
DISCOUNT RATE
&
HARBERffiR'S SOCIAL
IHE PRICING OF RISKY PROJECTS.
by IHIERRY F. BOLLIER
WP# 1500-83
MAY 1983
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
SOCIAL VPJJJNUION OF PROJECTS:
DISCOUNT RATE
&
HARBERffiR'S SOCIAL
THE PRICING OF RISKY PROJECTS.
by IHIERRY F. BOLLIER
WP# 1500-83
MAY 1983
cxxtments welcxmed
.,
.R
?
1992
I
ABSTRACT
In the presence of taxation
of private returns, should the government
attach to the gross cash-flows of a project the same value that the private
sector would have canputed when valuing the net-of-tax cash-flows?
One approach to this issue which has gained considerable following is
Harberger's Social Discount Rate (SDR) Theory.
In order to account for the
alleged tax distortions, Harberger and others propose the use of a weighted
average of observable market rates.
The rule is controversial in its treatment of fiscal
ignores risk.
effects
and
totally
The purpose of this paper is to develop a simple valuation rule that takes
into account both the tax distortions and project-risk considerations.
The
main result is that tax distortions might lead a hurdle rate for certainty
equivalent projects which differs from the market rate of interest, but
that in general the project risk adjustments are independent of the tax
distortions.
This rule is similar to the one proposed by Schmalensee[ 1976]
contradicts the highly referenced solution by Bailey and Jensen[1972].
,
but
I
would like to thank Don Lessard Stewart Myers, Richard Schmalensee,
Gerard Gennotte, Saman Majd and many other participants in the Finance
Workshop at the Sloan School of Management, M.I.T., for helpful comments on
earlier drafts.
I
am also grateful to David Laughton of the Corporate
Finance Division, Canada Department of Finance,
for his suggestions.
I
think
that I have greatly benefited from detailed and challenging
discussions with Fischer Black.
,
CONTENTS
Abstract
I
.
II.
Introduction
Social Discount Rate (SDR) Theories
5
A.
Harberger' 3 displacements
5
B.
Review of the results in the absence of project-risk
considerations
III.
The costs of Displacements - An Adjusted Present Value Approach..
A.
Adjusted Present Values in a world of certainty
B.
The Costs of Displacements in a world of uncertainty,
and the valuation of risky projects
IV.
V.
1
Systematic errors in Bailey and Jensen's Approach
Generalizations and Conclusion
Footnotes
Bibliography
14
18
19
27
37
45
HARBERGER'S SOCIAL
SOCIAL VALUATION OF PROJECTS:
DISCOUNT
RATE
AND
THE
PRICING OF RISKY PROJECTS.
I
.
INTRODUCTION
"For decades the most controversial issue in Cost-Benefit
Analysis has been the selection of the appropriate Rate of
Discount.
The controversy initially focused on the relative
merits of two candidate rates
(1) the (relatively low)
social rate of time preference, which we shall call the
consumption rate of interest (r) and which many writers have
identified with the after-tax rate of return on private
savings, and (2) the (relatively high) gross-of-tax rate of
return to privately financed investment, which we shall call
Recently, however, a nunber of
the investment rate (g).
analysts [ . ] have demonstrated that the Social Rate of
Discount (w) should be a weighted average of g and r. As a
the
result, the debate has been slightly refocussed,
candidate rates now being r and the weighted average w."
L.Sjasstad and D.Wisecarver[ 1977] 513.
:
.
.
,
The
valuation
theoretical
attempts
literature
dealing
with
public
to characterize what distinguishes public from private
assets and/or what makes the government a special investor when
the
behalf
extensive.
than,
nor
assets.
Discount
of
the
entire
The
cocnmunity.
list
of
The purpose of the current paper is neither to
to
propose
a
(SDR)
acting
on
controversies
is
settle
all
of
definite framework for correctly pricing public
Rather the focus will
Rate
project
be
on
one
approach,
Harberger's
Social
theory, as characterized in the above quotation from
Sjasstad and Wisecarver.
cost-benefit
the
This theory has gained considerable following
literature
recognition
with
along
in
extensive
and
application in government circles^.
Still, controversy remains on two specific grounds:
1)
the legitimacy
Harberger's
of
market
capital
alleged
externalities, which rationalize the computation of w.
2)
these
the treatment, in the public valuation
externalities
of
projects,
of
consistent with the accepted valuation procedures for
risky assets.
The treatment of uncertainty in a SDR framework
to
greatest
the
HarbergerC 1969/72],
traditionaly
of
deal
Sandmo
neglected
risk
Dreze[1971],
and
for
In
developed
Dreze[1974]
by
has
practice a key premise
the
between
wedge
a
theory
and
considerations^.
seems to be that the sole reason
gross-of-tax
in the private production sector and the net-of-tax market rate of
returns
time preference is a tax distortion.
among
The
confusion.
subject
is
academics
Canadian
Hood,Glenday
and
A lengthy argunent has been raging on
Burgess[1981],
Jenkins[ 1972], [1977],
(see
Evans[ 1981] ,[1982]
to
)
estimation of the Canadian discount rate;
settle
the
a rate
i.e.
question
that
is
on the
used
to
price any public asset independently of its risk^.
Whether uncertainty should justify price adjustments from
public
in an undistorted economy has also played a major role
perspective
in the debate about
developed
by
diversification.
Arrow
rates
social
Arrow
government-owned
a
and
of
Lind[1970],
portfolio
and
discount.
who
present
Che
extreme
view
was
point to the large size of the
an
argiment
of
perfect
Hirschleifer and Shapiro[ 1970] and Sandmo[1972] show that
and Lind's result is only a special case concerning the relationship
between private and public investments, and in general risk
However the discussion never integrates the
should enter the calculations.
developed
results
the
in
considerations
certainty
case
tax distortions and
the
for
Harberger's SDR.
One remarkable attempt to approach the issue is
referenced
article
by
government-owned-asset
perfect
Although most of the
M.Bailey and M.Jensen [1972].
contradicting
article was aimed at
Arrow
and
highly
the
Lind's[1970]
argunent
and thus centered on the
diversification,
relative abilities of the capital market and the government to share
they
present
Asset
Pricing
solution.
of
risk,
an exanple of desired risk adjustments, a Social Capital
as
which
Model
incorporates
Harberger's
average
weighted
Unfortunately they do not provide a proof or derivation of their
result.
The purpose of the current
valuation
rule
that
takes
into
project-risk considerations.
The
Jensen's
solution
is
wrong.
paper
account
both
analysis
to
is
the
show
will
derive
simple
a
tax distortions and
that
Bailey
The main result is that the tax distortions
might lead to define a hurdle rate for certainty equivalent projects
differs
from
and
which
the market rate of interest, but that in general the project
risk adjustments are independent of the tax distortions.
This
Schmalensee[1976]
.
solution
is
similar
to
the
one
Black[1981]
by
The main difference is that the current paper proposes
a very simple derivation of the result which relies on both
Fischer
proposed
Simple
Discounting
Rule
to
a
version
characterize
externalities, and an Adjusted Present Value Rule to account for
the
of
tax
these
in
public project valuations.
The following sections will attempt to deal with
the
issue
of puDiic project valuation at three levels:
1)
What questionCs) is being asked when the SDR is used?
2)
How can it be answered correctly?
3)
How can it be answered most usefully?
Many other aspects of public finance models will be
away
here.
(For
an
excellent
review
of those issues we encourage the
reader to refer to the recent book edited by R.Lind[ 1982]
In consequence
Section
2
discuss
I
discount rates.
the
the
paper
current
average
is
organized
literature
on
.)
as
follows.
In
Harberger-type social
The assumptions necessary for the issues to become neither
trivial nor untractable are also stressed.
weighted
assuned
discount
Results leading to
Harberger's
rate are sunmarized at the end of the section.
The simple valuation rule is developed in Section 3 and Section 4 shows how
Bailey and Jensen's social CAPM fails.
features
of
In the
concluding section the
main
the simple valuation rule are sunmarized and it is shown that
the sane valuation rule could be applied to more complex environments.
SOCIAL DISCOUNT RATE THEORIES.
II.
Harberger's displacements
A.
motivation
The
observations:
the
for
an
SDR
rooted
is
marginal before-tax-return on investment (the 'social'
return or marginal rate of transformation in the economy)
the
risk adjusted rate of time preference or, unambiguously
consunption
the
rate of time preference).
since apparently, if we were to set the
social
equal
rate
equilibriun
investor's
the
to
the
in
than
with
marginal
return
higher
certainty
This fact is disturbing
rate
preference
time
of
of time preference, rational
investment policy should induce the government to undertake
investments
larger
is
required rates of return on private savings (the investors'
after -tax
case,
empirical
in
than
all
available
common rate of time
the
preference.^
However the government should also
loss
future
in
benefits
take
into
account
the
due to the crowding out of private investments.
From a social perspective the opportunity cost of private investment is the
risk adjusted gross-of-tax rate of return in the private
effects
if
the
of government investment in a given industry were to crowd out the
sane amount of the private competing
incentive
for
industry,
then
transformation,
would
not
Hence the marginal
be
no
rate
the social rate of time preference, should become
the hurdle rate to be used in discounting future costs
public projects.
there
the government to undertake a project whose return is lower
than the private marginal rate of transformation.
of
sector:
and
benefits
from
SDR theorists point out that this line
only
reasoning
holds
limit where the elasticity of private sector investments per
the
in
of
dollar of public investment is
identically
contention
not necessarily true in a second best world
that
is
this
is
equal
to
minus
Their
one.
where the return on private capital is taxed and a wedge is created between
consunption rates of interest and marginal rate of transformation.
That the rules for optimal allocation
undistorted
economy
might
apply
not
of
resources
an
in
to distorted markets has been very
recently reiterated by Robert Lind[1982], in his introductory chapter of
symposiun
on
discount
social
rates:
"The problem is that the corporate
income tax and the personal income tax distort
depending
incentives....
Therefore,
where the resources for the public investment are drawn, the
on
criterion for acceptance differs and, more specifically, the discount
in
a
rate
one case should be the social rate of time preference, and in the other
case, it should be the marginal rate of return in the private sector ".
Allegedly any amount invested (B) by the government is drawn
either
from
investments
new
(I)
private
savings
(crowded out).
from
displaced
private
:
dS
1
and/or
This is formalized for the marginal dollar
invested in the following equation
(2.0)
(S)
dl
-
=
dB
dB
The social discount rate is then a "weighted average of
private
sector
marginal
productivity
of
capital
[i.e.
the before tax
return on capital ] and the net-of-tax yield on private savings ..[which
a]...
measure
of
the
marginal
sector" [Harberger 1969,1972].
rate
the
is
of time preference in the private
dl
dS
w
(2.1)
=
i
(1-T.)
^
-
i
(1/(1-T„))
dB
dB
where:
is the observed (riskless) market rate of interest
L is the marginal tax rate on savings
T^ is the marginal corporate tax rate.
i
Equation
Harberger's
is
(2.0)
fundamental
displacement
equation.
Detractors of Harberger's social discount rate argue against
the theory on three specific grounds
1)
:
Equation (2.0) does not make much sense from the point of
view of a social welfare maximizing central planner.
to
the
one
familiar
optimal/neutral tax literature.
the
from
context Diamond and Mirrlees[ 1971] and Auerbach[ 1982] in
analyses,
recommend
decisions
which
use
the
equate
of
rates
discount
public
the
similar
The issue is
and
for
private
their
that
In
respective
public investment
marginal
rate
of
that
the
transformation.
production
Aggregate
government
efficiency
requires
both
chooses an optimal level of equilibrium interest rate (at which
the private sector will discount after tax project cash-flows) and that
designs
its
tax
transformation
equilibrium
exactly a unit
require
can
of
across
equal
are
it
incentives
and
be
shown
private
schemes such that the marginal rates of
activities.
productive
In
investment^.
Hence
1
,
optimal
an
would
taxation
and the right decision rule
must assune this identity when defining the hurdle rate to be used
In
such
that a unit of public investment displaces
(-dl/dB) to be identically equal to
public valuation.
it
in
the
this event the weighted average rate (2.1) boils down
identically
being
private
the
sector
(gross)
marginal productivity of
capital, hence:
Proposition
public
sector
does, i.e.
should
1
except
:
value
for
projects
cash
flow
externalities
in the same way the private sector
the present value calculation should be performed either on the
total cash flows discounted at the before tax marginal rate
capital
the
or
equivalently
sector discount rate
of
return
on
on simulated after tax cash flows at the private
.
The proposition is the only one acceptable in an undistorted
(first best world) economy.
ability
Or it assunes that a central planner
has
the
to enforce Diamond-Mirrlees standard of production efficiency.
either case the interest
consumption
levels
rate
level
(after
tax)
is
chosen
In
optimally,
are fixed and a unit of public investment must exactly
displace one unit of private investment.
But Diamond and Mirr lees' [1971]
between
flows
result
that
all
In
fact
that
the
the household and production sectors may be taxed.
this equilibrium will be socially desirable only to
central
requires
the
extent
planner does have sufficient control of the necessary instrunents.
If the government cannot monitor continuously the instrunents
which
would
allow him to equalize the marginal rates of transformation and optimize the
of interest rate, and if a tax on capital income has created a wedge
level
between the rate of time preference and the rate of transformation
tax
rates)
in
(before
the economy, there does not seem to be a case for assuning
such an identity (-dI/dB=1) in public valuations.
For the identity to hold
started
from
we
need
both
that
the
economy
production efficiency (optimal interest rate and equal rates
of transformation)
,
and that the private sector anticipates that,
for
any
shift
equilibriun
in
the government will take the necessary
conditions,
actions to bring back production efficiency in the economy^.
In the distorted and non
private
investors
equalize
the
marginal rate of time preference.
effectively
the
consumption.
In
shadow
other words
between
allocation
production-efficient
Hence the after tax rate of interest
for
the
lower
marginal
any
amount
determines
rate
and savings.
equilibriun
the
after
of
rate
tax
of
the
is
forgone
investors'
The same investors would be
better off if they were allowed to invest in any investment
than
the
after tax rate of return on capital with
price
consunption
economy
yielding
more
Except in the very
interest.
unlikely event where consumption is inelastic with respect to the
rate
of
interest, shifts in savings are bound to occur.
Hence a public project would tend to displace
consunption
and investments.
average
of
the
of
marginal
transformation
of
the
public
project.
Since
better
a
in
the
economy,
represent
the
opportunity
dominant strategy for the public
authority would be to eliminate distortions due to taxation and move to
undistorting
set
an
of transfers in order to achieve at least the same level
of social utility, we must assune then that
available,
a
of transformation and the lower
rate
marginal rate of time preference might
cost
private
If the central planner's objective is not to
control for equalization of rates
weighted
both
such
opportunity
an
is
not
or for some unspecified reason actual taxation does not lead to
production efficiency.
Under the alternative assunption that
not
achieve
not
incompatible
assunptions
production
on
with
the
government
does
efficiency, Harberger's alleged displacements are
general
equilibriun,
nor
do
they
violate
the maximization behavior of the private sector.
any
The only
10
requirement is that the economy be distorted by the taxation of returns.
The second line
2)
of
criticism,
stressed
as
in
modern
finance theory, is that mistakes are often made in attempting to derive the
opportinity
financing.
of
cost
a
each
For
project-specific
given project from the apparent costs of explicit
particular
project
there
might
exist
a
cost of capital which can be best estimated by looking at
similar competing projects rather than trying to trace the required rate of
return from the different securities used for its financing.
The "natural" temptation for the financial economist
that
argue
the
project
to
marginal rate of return on private and public investments
should be the same in all events.
public
is
No matter what
might be the impact of
a
on private consumption and investment the opportunity cost
is represented by the marginal private investment
the goverrment
:
should
never invest in projects with lower returns.
Proposition
1
can be
rewritten
as
despite
follows:
any
actual effect on private consumption the public project should be valued at
the
before tax marginal rate of return on investment (as if the elasticity
of private investment was identically equal to
decision
one)
.
Again
the
correct
rule is obtained if one always assunes that the public project is
crowding out the same amount of private investment.
This line of argument would surely be correct if the
to
undertake
the
marginal private investment was open to the government.
This might not be the case in reality
:
private sector businesses may have
a competitive advantage over public enterprises for all
and
the
option
production
frcm the private one.
sorts
of
reasons
function of the public sector might be very different
Other reasons could be outright bans
on
government
11
involvement
specific
in
nunerous
and
of
sectors
economy, or the
the
uninforceabiiity of an efficient set of subsidies to the private sector.'
We shall assune in what follows that the private and
sectors
choose
their
level
of
investment
from
two
public
different sets of
technologies".
3)
goes
Harberger,
A third line of argument, which Lind[1982] attributes
as follows:
to
public expenditure contributes to the budget
The financing of the additional
deficit and must be financed through debt.
private
anount of debt will crowd out an equal amount of
investment
(and
again we would be lead to assune -dI/dB=1).
We must accept the
expected
public expenditures.
intervals
and
the
system
tax
cannot
be
flexible enough to absorb any fluctuation in the level of
be
to
that
fact
the
Taxation
marginal
can
only
dollar
be
adjusted
at
discrete
time
public investment will always be
of
financed out of government debt.
However the most convenient stand with respect to this issue
is to adopt R.Barro's[1974] counterargument
government's use of debt which is just a
taxation.
future
:
agents are not fooled by the
current
taxation
for
The debt issue is fully transparent in that sense.
The
shift
of
allocation between consonption and saving should be the one that would have
prevailed if first period consunption had been
debt
issue
taxed
instead
(i.e.
the
should not affect savings compared to the no-debt case and all
additional funds are taken from consumption)
Before concluding
equation
(2.0),
we
can
this
chapter
on
make one last statement.
the
rationale
behind
It is true that if the
12
government
action
of
financing
were
the
not
project
fully
transparent
Barro's
in
private
of
the
the
economy.
not
states
He
resources
are
currently
danand
adjustments
and
,
affected
when
the
induced
of
reallocation
effects
would
be
inexistent
favors
capital
clean
labor,
could
or
be
in
also
be
better
a
although
air),
ignored if production
efficiency was allowed, as in Diamond and Mirrlees[1971].
concludes
could
savings
The level of
utilization of factors of production (e.g.,
these
increase
be made at the expense of minor
will
crowding out in the private sector.
fully
that an increase in production of
unemployed resources, if they exist, might very well match the
aggregate
private
original public project and/or its financing
might very well depend on whether or
in
more
Cn the other hand as pointed out in Lind[1982], the impact on
investments
employed
the
result in an additional increase in the
could
interest rate which in turn would contribute to crowding out
investments.
sense,
Eventually
Lind
:
First, one cannot reasonably divorce the analysis of
public investments from macroeconomic considerations because the
financing of public investments has different impacts on the economy
depending on the state of the economy.... Second.... the choice of
criteria by which to evaluate public investments in a second best
world depends critically on how one beleives the economic and the
political systans work....
In
displacement
sunmary we
equation
(2.0).
state
can
a
set
for
the
opportunity
cost
of
public
.
1)
taxation
assunptions
The same assumptions support including the
alleged displacements in the analysis of the
investment
of
the
economy
is
distorted
of return and hence in equilibriun
by
a
and
personal
wedge is created between the
marginal rate of transformation in the economy and
time preference in the private sector.
corporate
the
marginal
rate
of
13
2)
the government does not achieve production efficiency and
does not fully control the level of interest rate.
3)
efficiently
the
the
can
government
marginal
private
neither
undertake
nor
subsidize
sector investment and hence public and
private sectors optimize over different production sets.
M)
in excess
government borrowing activity does not absorb any savings
of the level that
vgould
have prevailed if the
directly financed through current taxation.
project
had
been
14
B.
Review of results in the absence of project-risk considerations.
assumptions,
Harberger's displacements equation (2.0) must be incorporated
in any analysis of the opportunity cost
equation
is
certain
given
We have argued in the previous section that,
of
This
investments.
government
general, but we can expect the displacements of both
totally
saving and private investments to be closely related to the characteristics
of the project, e.g., project total return, project life, type of resources
used in production, etc.
Sandrao and Dreze[1971] and
need
point
Dreze[1974]
out
the
of a detailed specific assessment of the "opportunity cost applicable
to investments at a particular time and place"
.
Nevertheless in estimating
equation (2.1) most empirical studies have assuned that there are
project-specific)
(non
Evans[ 1981, 1982]).
standard
adjustjnents of private savings and investments per
dollar invested (see JenkinsL 1972, 1977], Burgess[1981]
and
for
The
account for the "financing"
social
discount
externalities,
rate
is "designed" to
(w)
independent
Glenday
HDod,
and
,
of
the
specific
project under consideration.
Although this assunption may be reasonable in the derivation
of theoretical
displacements
models
in
aimed
at
studying
the
bias
be
used
the
the opportunity cost of a project, the simplification is
often improper when the purpose is to derive a schedule of
to
by
introduced
in actual valuations.
discount
rates
We shall come back to this issue later
when we will discuss the possible implementation of the valuation fraraevork
in real life cases.
For now we shall make the same assunption.
The main SDR result remains Harberger's[ 1969, 1972]
(2.1).
equation
Most of the more recent papers have proved Harberger's proposition
in two period settings (Sandmo and Dreze[1971]) or in multiperiod
settings
15
(Dreze[1974])
reconciled previous approaches with the weighted average
or
solution (Sjaastad and Wisecarver[1977])^'
certainty
the
In
case
opportunity
social
the
cost
of
capital, w, can be rewritten in a notation which has become standard in the
literature^^
:
w
(2.1)'
where
and
=
dS/dB
-
dl/dB
g
is the after-tax indirectly observed rate of time preference,
r
is
g
r
before-corporate-tax
the
rate,
indirectly
or
observed
marginal productivity of capital.
The
expected
signs
respectively.
government
of
dS/dB
These terms express
borrowing,
B, upon
dl/dB
and
net
the
are
positive
of
impact
an
and
negative
increment
of
private savings, S, and private investment,
I.
Harberger argues that the model
multi-market
risky
economy,
derives the "general" formula for w
w
=
E i^^.(1
- T^i^)
T^^^
is
to
a
investments
and
securities.
He
:
dV^B
k
where
generalized
be
where required rates of return and tax rates
can differ across classes of investors,
(2.2)
can
-
E
ij/(1 - Tj,j) dIj/dB
j
the marginal rate of personal income tax applicable to
the k^" class of savings, and T„
incone tax applicable to the
j
•
is the
marginal
rate
class of investments.
of
corporate
16
In
sub-markets
(enough
are
displacements
(2.2)
sunmed
over
all
possible
sub-markets are defined for each to be homogeneous in
terms of required returns and marginal tax-rates)
In
w
(2.3)
the standard notation,
zr^^dSi^/dB
=
igjdIj/dB
-
k
j
The partitioning is justified intuitively because investment
sub-markets could be
different
marginal
thought
systematic
tax
tax-transfer
rates
rules)
risks,
of
different
income
on
and,
different
as
cost
capital
of
government
(the
equivalently,
sectors
industrial
might
and
different
different
set
different
because
with
income-class
citizens face different marginal tax rates on their security holdings.
Harberger[ 1969, 1972] correctly notes that each "i. [and
obviously
incorporates a premium for risk and whatever other factors which
influence portfolio decisions.
Dj
i^^]
They can be represented as i-=i^b^,
is the (quite possibly variable)
in the jth
activity
and
i^
is
where
"risk premiun" required for investment
(risk-free)
the
rate
of
interest
on
except
to
government bonds"
We shall not review
stress
that
in
theories
SDR
domestic
:
The measuranent of w based on estimated
investments,
elasticities
of
displaced
elasticities
consunption
of
and
the
Although not always stated explicitly, they use w as
the
incremental foreign financing^
2)
longer,
practice social discounters intend to perform two related
exercises with Harberger's social discount rate
1)
any
^.
17
discount
sole
rate
any public project valuation, independent of its
for
risk.
Although w in (2.3) corresponds to
of
capital
average
weighted
cost
national portfolio, it is inappropriate to use it to
the
for
a
allocate resources among the assets of this portfolio when potentially each
project contributes differently to the total risk.
This is
fundamental
a
principle of financial economics.
As Harberger recognizes it, the proposed w incorporates
premiuns
hence w calculated from (2.3) could
of each activity;
risk
for
the
neither be viewed as the certainty equivalent rate that could
used
be
to
discount correctly the certainty equivalent cash-flows of the projects, nor
as
a
risk
discount
adjusted
that follows should
analysis
The
rate.
demonstrate these points.
proceeding
Before
warning.
Whether
valuation
the
displacements is a separate
section
of
sunmary
a
the
further
let
account
should
issue.
We
arguments
us
state
for
presented
have
against
schedule
is
necessarily
though,
likely
mean
to
to
recognize
diverge
from
that
in
[of]
in
hypothesized
the
current
As
pointed
out
by
principle the optimal tax
production
efficiency,
does
not
that we have other choices in practice since it might be
"extranely difficult even in a very simple model to
direction
the
the production efficiency
hypothesis and we will not extend on it any further.
Auerbach[1981]
additional
an
this
divergence
...
".
Hence
difficult in practice to estimate the displacements.
calculate
it
merely
the
might be extremily
18
III.
THE COSTS UF DISPLACEMENTS - AN ADJUSTED PRESENT VALUE APPROACH
Let me first restate the problem.
Under
investment
will
be
the
stated
conditions
financed
in
public
the
How can we best account
reflects
the
social
opportunity
that
clai^i
of
project
a
value
This is simply:
can always be adjusted for potential externalities.
it
Stiglitz[1982]
cost of capital.
reminds us that there is a trivial sense in which the value
all
these
for
The social discount
valuation of projects?
rate theory has proposed the weighted average w, under the
truly
public
the
2
out of displaced savings and investments as
described by Harberger (equation 2.0).
displacements
Section
in
future costs and benefits by use of the marginal (social) rate of time
The public valuation differs from a private valuation
preference.
to
the
extend that the total project cash-flows differ from the direct cash-flows,
appropriable
in the private sector.
Because of the wedge between the rate
of time preference and the rate of transformation cash-flows of the forgone
private investments crowded out by the new project
part
of
the cost externalities.
externalities;
as
The total cash-flow approach would allow
us to value sequentially the direct
cost-benefit
treated
be
should
here
cash-flows
the
of
present
the
project
of
value
and
the
the
forgone
investments.
Sjaastad and
Wisecarver[1977]
have
argued
approaches are identical if consistent assunptions are made.
they
that
In
the
two
particular
were dealing with the case of certainty where no discrepancies in the
rates entering w could be attributed
uncertainty
solution.
no
simple
adjustment
to
could
risk
premia.
reproduce
The total cost-benefits approach offers an
In
the
case
of
the weighted average
alternative
way
to
19
study
problem
the
be very easily accomodated in the framework
can
that
familiar from the finance literature, known as the adjusted
(APV)
of
project.
a
potential
only
The
problem
value
present
provide the
to
is
appropriate value for the shadow price of forgone investments,
In
approach
what follows
solve
to
problem
the
propose
we
of
the
use
the
of
alternative
public valuation in the case of
the
uncertainty.
shall
We
derive
first
the
one
case.
period
We will proceed
instructive to examine that simple problem in some detail.
in two steps
A.
is
It
:
We first introduce the adjusted present value approach in a world of
part
We introduce the APV approach in this simple framework in
certainty.
show how the rule can be used to isolate projects which would have been
to
private
rejected as part of the
projects
sector
opportunity
i.e.,
set;
public
which display positive present values only because of the alleged
displacements.
B.
Second we derive a fundamental result about
investments
in
a
world
uncertainty.
of
This
the
value
of
along
result,
restatement of the APV rule, will form the basis of generalization
of
case
valuation
multi-beta
projects,
risky
procedures
CAPM
for
economies)
the
to
the
,
multi-period
underlying
forgone
with
to
a
the
case and to more general
investment
(real
options,
as briefly discussed in the conclusion of the
paper.
A.
Adjusted Present Value approach in a world of certainty:
Since we want to consider the effect of tax
distortions
on
the valuation of public investments we introduce a tax on corporate returns
20
Tq
j
tax
This
.
rate
For simplicity we
vary across industries (j).
can
assune as a first step that there are no personal taxes on savings
in
competitive
equilibrium
private
the
sector
equates
project
after-tax-return on the marginal dollar invested in a
with
the
Hence
.
expected
(i^d-T^ J)
the required rate of return on savings (r.) for a comparable level of
systematic risk.
Under these assunptions we want to determine the current net
present value of a proposed
end-of-period
risky
cash outflow Cq.
hence,
net
single
period
project
will
that
bring
an
cash inflow of C^ and will require a current net
the
jhe project is to be undertaken by
government
is drawn from the public investment opportunity set.
and,
(We assune in
accessible
to
The rationale for the adjusted present value approach is
to
particular that the marginal private investment are neither
the public authority, nor can be efficiently subsidized.)
disregard
adjustment
any
of
the
discount
rate
and
value
project-specific cash flows and cash flow externalities at the
social
discount
Arbitrarily
rate.
we
set
the
social
all
the
appropriate
rate
of
preference to the after-tax-rate of return in the private sector, hence
time
we
will use this (lower) rate to perform the present value calculations.
In this simple case where the only cash
correspond to the alleged displacements, we can write:
(3.0)
NAPVp^^^cc^)
=
NPVb(Ci) + NPVf(Ci)
=
PVjjCC^) - Cq + Cq -
PV(Hp
flow
externalities
21
The first part of the public valuation (NPVjj(c-i) = PVt,(Ci)-Co)
project.
is equal to the value of direct cash-flows from the
part
(NPV^(C^)
-
Cq
=
PV(H-|)
)
corresponds
the present value of
to
displaced future consunption (H^) due to the displacements of
private sector investments.
we
second
The
savings
and
By analogy to the corporate sector environment
shall refer to this second term as the present value of the "financing"
If the new project
externalities.
financing
would
preference,
marginal rate of time
entirely
financed
out
of
value of forgone consumption when discounted at the
present
the
savings,
be
to
is
would
externality
be
be
exactly
equal
to
Cq,
the
identically equal to zero and the public
present
value of the project would be equal to the
value
of
the
direct
As we shall see below in general this will not be the case.
cash- flows.
Since we are in a world of certainty, we have the
present value in the simple formulation
base-case
:
^1
(3.1)
PVt^(Ci)
=
1
+ r^
where, r^ is the riskless rate of interest, assuned to be equal to the
social rate of time preference.
Given the sole taxation of corporate
returns
equation
and
for the displacements we have the following expression for the total
(2.0)
period
amount of consumption displaced (see below) by the new project next
(H^)
:
H^
=
[
I
k
(
Ur;^
)
dS;^/dB
-
^(U
Tj/d-Tcj)) dIj/dB
]
.
Cg
J
The investment side, i.e.
the
right-hand
sum,
represents
the social opportunity cost, in tenns of future forgone consumption, of the
22
of
amount
capital
diverted from private sector production.
discount rate theories, it corresponds to
respective
sectors
plus
left-hand
sura
savings
in
investors;
of
future
return
transformation)
in
which
cash- flows
diverted
of
cost
accruing
of
incremental
the
to
tax
is equal to the after-tax return on savings (one plus the
of
rate
the
On the saving side, the
corresponds to the social opportunity
terms
tax
i.e., the marginal return on savings minus the incremental
revenues,
marginal
of
private sector use.
alternative
their
in
after
i.e., the marginal before-tax
the tax revenues;
productivity (one plus the marginal rate
funds
the
As in social
time
preference)
private
the
in
sector.
(See
HarbergerC 1969, 1972] for a formal description of these displacements).
incremental
the
If
project
the
small,
is
effect
on
equilibrium prices will be trivial, so that H^ can be valued using marginal
returns.
In equilibriun all
market rates of
must
return
equal
r^,
hence we have:
(3-2)
H^
=
Cq +
[
ETf dSi^/dB -
Z
k
j
dIj/dB
Tf/d-Tcj)
]
Cq
since
EdS^/dB
or
we
can
rewrite
H^
by
-
Edl./dB
introducing
=
1
an adjustment variable (T) which
involves only the tax rates and the respective elasticities of
investment,
(3.3)
H^
=
CQ(l+rf.T)
saving
and
23
where
T=
(3-4)
zdS^/dB
-
z
K
(l/Cl-T^j)) dIj/dB
J
Notice that if the elasticities have the
and
at
one T^-
least
> o,
then T
>
anticipated
signs
and in general T< 1/(1-1..) (if the
1,
tax structure is not too heterogeneous)
Hence in the certainty case the APV calculation collapses to
a rather trivial exercise
:
Cgd+r^T)
C-|
NAPV
(3.5)
j^(Ci)
- Co + Co -
=
1+r^
l+r^.
equivalent
We can notice that the decision rule is
one
advocated
by
social discount rate theorists, which states
social discount rate,
=
-— -
the
use the
to value public project cash-flows.
rr-j,
Ci - Co(l+rfT)
Ci
^PW(Ci)
:
to
-
Co
Clearly NAPV
^^
-—
=
will
and NPYgQ^
signs, although not the sane value.
In
always
display
the
same
any case the adjusted present value
approach (4.5) does not complicate the valuation beyond the steps that were
requested in the adjusted discount rate approach.
The remaining of the section
discuss
several
features
on
of the APV approach.
the
In
certainty
case
will
particular we show how
the valuation behaves under the assunption that public investments displace
one for
one
private
investments
(the
optimal-tax/production-efficiency
24
and
assumption)
display
a
how
the
can be used to isolate the projects which
rule
positive NPV only because of favorable hypothesized displacements
of savings.
We can remember from the discussion in Section
link
between
sensitive issue.
Some would like to argue that -dI^/dB=1
identically, so that r^x collapses into
an
to
across
equal
crowded
investment
efficiency
out;
which
sectors
>*£«/( l-T^,^,)
here
that
Notice
.
assunption;
or
or
would
that
be
in
rate
transformation
of
of
equivalently
with
consistent
production
the
the marginal rate of
instance
any
discount rate equals '"f/CI-TQ^))
the
or proposition
1
(Section 2)
:
cost
In other worlds
.
neo-classical
these critics state what we will refer to, hereafter, as the
rule ^^
the
we could assune that tax rates are
transformation in the private sector is the only relevant opportunity
(and
we
class "c" such that the before-tax-discount rate
investment
does actually correspond to the marginal
private
the
the alleged displacements and the opportunity cost of public
investflient is the
refer
that
2
take the public project if and only
if
NPV^^(Ci)
-
=
Co
>
l+^f/d-Tcc)
Although we have discussed at length the
assunptions
which
have
lead
us
to
reject
of
set
principle
in
realistic
this extreme
position, we would like to keep them as a benchmark in the public decision.
We can adapt the valuation rule (3.4) and (3.5), in
the
a
order
to
distinguish
proportion of the public value that would have been obtained by use of
neo-classical rule
and
the
hypothesized shift in savings.
proportion
that
accrues
because
of
the
25
^^
This can easily be done by transforming (3- 4) and (3.5)
-
zdSj^/dB
=
(3.7) T'
k
z(1/(1-T^j)) dIj/dB
-
(1/( l-T^^,)) (dl^^/dS
+
1
J
1/(1-Tan)
==>
Co TfT'
Co(Urf/1-Tcc)
Ci
(3.8) NAPVp^^^(Ci)
Co ^ Co -
=
1+r^
I+Pj
1+r£.
(3-8)
would
decision rule as (3-6), the neo-classical rule;
i.e.,
Clearly if we were to leave out the last term,
lead
same
the
to
NPV^^(C-,) and NAPVpub(Ci) + Cq
rfTVI+rf would be both positive or negative
Hence by analogy we rewrite
simultaneously.
(3.9) NAPVp^^^(Ci)
:
NAPV^^c^Ct) - PV(H'i)
=
where,
^1
NAPV^^(Ci)
Cod+Tf/I-Tcc)
=
l+r^
H'
1
=
Co rfT-
NAPV^^(Ci)
magnitude
1+r^
since
^^
'^P^nc(Cl)
^^^^
^"
general
have
different
the denominators are not the same, but as discussed above
the signs will always be identical.
Notice also that
H'
,
the case for two possible sets of reasons
First, - dI./<jB <
1
This
will in general be negative.
is
:
and a tax structure of corporate returns
not too heterogeneous imply, as pointed
out
before,
1
<T< 1 / 1 -T^^,
;
hence
26
T'<0.
Second,
(neo-classical
in
the
case
but
the
taxation
even
assunption)
where
of
saving
no
corporate
disparate, and the public investment tends to crowd out low
projects
(relative
to
T^^,),
might again be negative.
However the converse can
also
returns
taxed
would tend to be close to
T
created
is
private
and hence T'
1
be
is
true
if
the
types
of
public investment tends to crowd out highly taxed private projects.
consequence
In
will
we
tend
to
have
two
positive-net-present-value public projects:
1.
Type one
which
projects
elements in (3-9) are positive;
display
positive
NAPV
^^
because
both
i.e., the project would have been accepted
even if we were using the neo-classical assumption.
2.
size
Type two projects which display positive NAPV
^^
because
only
the
of the externality (the present value of the alleged displacements in
excess of the
neo-classical
assunption)
is
large
enough
offset
to
a
negative net present value under the neo-classical assunption.
We might want to isolate type two projects
the
theory
because
nor empirical evidence can be expected to provide the decision
maker with reliable estimates of the displacement equation(T or
framework
neither
allows
easily
to
pin-point
T').
The
such projects, to proceed to some
sensitivity analysis of the estimate of T (or T'), and at the same time
to
keep as a benchmark the neo-classical valuation.
We can stress an additional feature of (3.9).
reinterpret
the
last
period) necessary to get
desirable project.
term
a
Provided
we
we can compute the "bribe" (tax break in next
private
investor
to
undertake
the
socialy
27
B.
The costs of displacements in a world of uncertainty, and the valuation
of risky projects:
I
do not intend
restate
to
the
Bailey
and
Jensen[1972]
Their article
argument showing why risk should enter into the calculation.
brought an end to the general acceptance of the use of the
have
to
seetns
riskless rate in public valuations (at least on the grounds of the
diversification
argument)
.
Much
of
perfect
discussion was centered on the
the
relative abilities of the capital market and the government to share
the
Along
same
each
economy
lines
consuner
diversification,
Sandrao[1972] formally shows that in a stock market
has
unlimited
possibilities
portfolio
for
all risks priced in the market are true risks to society,
the sources of which are to be found
Hence
risk.
in
the
nature
of
the
technology.
the government has no advantage over the individual consumer in this
respect.
The most prominent feature of these approaches is
only
public discount rates should contain risk premia, but also that these
that
should correspond to the one used in the private sector.
to
not
be
found
also
in
Schmalensee[1976]
This
feature
is
when he derives the result with
complete insurance markets.
We shall share the same point of view and
assume
that
the
public and private sectors are identical in their assessment and pricing of
risk.
(For
most
of
the
following discussion, risk means the project's
marginal contribution to the variance of the
i.e.,
systematic
risk
in
a CAPM sense, the
cash-flows with the return on the market.)
aggregate
market
portfolio;
covariance of future project
28
Public valuation under uncertainty
:
As introduced in the certainty
case
propose
we
to
value
the total costs and benefits of the project, inclusive of the
sequentially
induced costs of forgone investments.
There is a trivial sense in which we can always apply a rule
equivalent
chapter,
previous
as developed in the
cash-flows.
main
The
once
it
apply
we
certainty
to
result in the next section is that the
calculation of certainty equivalent displacements can also be simplified to
a
trivial
equivalent
once
exercise
of
we
realize
that,
in
certainty
equilibrium,
market rates of return must be equal to the riskless
risky
rate.
now
us
Let
restate
our
problem
original
risky
a
in
environment.
In
traditional
the one period setting the necessary conditions
Sharpe-Lintner
for
the
capital asset pricing model (CAPM) are assimed
to hold.
As before we want to consider the effect
on
the
valuation
public
of
corporate returns T
Now this tax rate can vary
distortions
we introduce a tax on
Hence
investments.
of tax
across
industries
and
risk classes (again for simplicity we assune as a first step that there are
no
personal
taxes
savings)
on
.
Hence
in
competitive equilibrium the
private sector equates the expected after-tax-return on the marginal dollar
invested in a project
(Ep,i •(l-T_-i))
cj
u J
with the required
rate
of
return
on
savings (Egrp for a comparable level of systematic risk.
Under these assunptions we want to determine the current net
present value of a proposed
end-of-period
risky
net
single
period
cash inflow of
project
C- and
that
will
bring
an
will require a current net
29
cash outflow Cq^
government
before
as
the
project
is
undertaken
be
to
the
and, hence, is drawn from the public investment opportunity set
The public remains the residual claimant on the government assets
excess
by
return
(or
deficit)
period domestic consunption
is transferred to(or taxed away from)
Although
.
and
there
might
be
a
any
end of
distinct
debt
or tax-transfer payments, the net position of the public is as
transaction
if the government had issued equity on the new asset.
Again we set the social discount rate to
after-tax-rate
of
return
in
be
equal
to
Since we want to use a
the private sector.
single riskless discount rate we must also adjust all cash flows for
derive
i.e.
the
certainty
the
risk;
equivalent cash flows, which is an easy task
under the conditions of the one period CAPM.
Formaly we can write in our simple case where the only
flow externalities correspond to the displacements
(3.10)
where,
NAPVp^^jCC^)
NPV^^
is
the
cash
:
=
NPVb(Ci) + NPVf.(Ci)
=
PV^j(Ci) - Cq + Cq - PV(Ht)
base case net present value of the direct cash flows
fran the public project,
^P^^ is the net present value of the 'financing'
the
present
value
of
displaced
externalities,
i.e.
future consunption (H-) due to the
alleged displacements of saving and private sector investments.
Since CAPM holds, we have the base-case present value in the
following certainty equivalent formulation
:
30
(3.11)
PV,(Ci)
CE^^l)
11
+ r^
=
E(Ct) - h.COVCCi.PM)
1
+ Pf
where
h is the market price of risk
Tf is the riskless rate of interest
is the after-tax-rate of return on the market.
^l/[
Given the taxation of corporate returns and
Harberger's
for
displacements
have
we
total anount displaced next period (H.)
=
Hi
[
E(
1+rj^
dVdB
)
-
(2.0)
the following expression for the
;
Z
k
equation
(
ryd-T^^j
1+
dIj/dB
)
]
.
Cq
j
Notice that the returns entering H, are random variables, so
that H- is also a random variable.
(H^
expected displacements
with the return on the
practice.
There
we might want to calculate the covariance of H^
market
is
order to assess the present value of
In
portfolio.
much
a
This
way
simpler
might
difficult
be
value
to
the
in
uncertain
displacements, which we introduce now.
H-]
simplifies to
(3-12)
H^
=
Cq +
irjt
k
[
dVdB
Eryd-Tcj)
-
dIj/dB
]
Cq
j
since
Ed\/dB
zdlj/dB
-
k
=
1
j
or we can rewrite H^ in a form actually proposed in Harberger[ 1969, 1972]
(3.13) H^
=
CoCUrf.T)
+
[
Lir^-r^) dSj^/dB
k
-e
(rj-rf)/( l-T^j)
j
wnere,
(3.14)
T=
zdV^fi
k
-
zd/d-T^j)) dIj/dB
j
dIj/dB
]
Cq
il
Notice again that if the elasticities have
signs and at least one T
>
then T
o,
of
derivation
The
displacements
.
>
the
to zero
of
value
present
are
(3.13)
alleged
the
a
function
of
excess
.
a
obvious.
A(r^_r^)
Let
,
this
result
The reason, here, is actually quite
private
where r^ is the return on an asset valued in the
satisfies
consequence
Fischer
see
the present value of a particular cash flow
consider
us
of
discussion
general
Black' s[ 1981] Simple Discounting Rule.
in
1.
and hence the present value of those returns must be equal
returns
For
and
anticipated
can be simplified if we recognize that the cash flows in the
second term of the right hand side of
market
the
the
CAPM, and A is constant.
:
market
The present
value of this particular cash flow is
A[(E(r )_r^) _ h.COVCr^.rM)]
PV(A(r^.rf.))
i
=
+ r^
1
But since we can replace E(r^)
have
r^ +
=
B(,(r[«j-rf)
=
r^ + h
cov(rj,,r|vi)
:
A[r^ + h.covCr^j.r^) - r^ - h.^VCr^.r^)]
=
PV(A(r^.rf.))
1
+
r.
Hence we have if CAPM holds
PV(H^)
(3.15)
Cq
=
NPVf(Ci)
=
(l+rf.T) / (1+rf.)
Co - PV(Hi)
=
and the total value of the project
CE(Cp
^PVb(Ci)
=
_
1+r^
and
Co rf(1-T)/(Urf)
:
Co(UrrT)
-—
,
,
we
32
characteristic
Oie
adjustment
of
this
result
that
is
and the tax adjustment are independent of each other.
the pricing of the
displacements
risk
of
neither
is
savings
affected
Moreover
nor
displacements could be performed exactly as in the certainty case
using
the
riskless
rate of interest and the tax variable
independent of the risk preraia embodied in the
different
investments
with
displaced.
This
result
the
by
while the adjustment for the
investments;
and
taxation
by
risk
the
agrees
T,
by
just
and hence is
returns
on
the
Schmalensee[1976] but
contradicts the highly referenced solution by Bailey and Jensen[1972].
As
shall see in part IV, their solution proposes the use of the variable T
we
to adjust both the riskless rate and the risk premia, which is incompatible
with our result.
If we consider the complete Harberger world with personal taxes on security
returns we can derive a similar result.
market
not
will
the
be
right
Now
measures
the
rates
observed
the
in
the riskless rate of time
for
preference and its risk adjusted counter-parts.
If if, i^ are the market rates and we assune a flat personal
tax rate T^
^
then we have a CAPM equilibriun either in terms
of
observed
market returns (i), or in terms of after-tax-returns r=i(l-T^)^^
Then the base-case NPV is
NPV^
=
[
(
ECC^) - h.COVCC^.rM)
(Ur^)
]
- Cg
ij/d-T^c)
^
dIj/dB
)
/
and the displacement effects are
H^
=
[
j;(
k
1+
i^d-Ti)
)
dVdB
-
E
j
(
1+
]
.
Cq
33
Repeating the transformation
U
H^=[
(ifxT) +
E
dV^B
(i|^-if)(1-Ti)
-
k
((ij-if)/(1-Tcj)) dIj/dB
Z
]
Cq
j
where (now adjusted for both personal and corporate taxes)
T
=
dV^B
l^'^-'^i)
-
j(
/(1-Tcj)) dIj/dB
1
and as before the present value of the excess returns equals zero.
PV(H^)
Cq
^
1+ (ifxT)
(
.
(Urf)
/
)
Hence
and
NPV^
(3.16)
=
=
Cq
NAPVp^^(Ci)
[
(
.
=
-
1
(
I^PVb
E(C^) _ h.Cov(cT,rM)
)
(
-.
(1+(if.T)) / (1+rf)
)
)
NPVf
/ (1+rf)
]
-
[
Cq
(
1
In short our net public adjusted present
+ ifT
)
value
(l+r^)
/
equals
]
the
base-case NPV plus the present value of future taxes displaced (computed as
if
all
were equal to the riskless rate r^ and discounted back at
returns
the riskless rate (i.e.
As in the
solution
into
an
Fischer Black's simple discounting rule).
certainty
case
=
1)
:
can
transform
the
general
equation that separates the neo-classical valuation and
the value of the displacements in excess of
(-dl^/dB
we
the
neo-classical
assunption
34
(3.17)
NAPVp^j^(cT) = NPVb(CT)
+
Cq if( 1-Ti-T)/( Ur^)
<==>
(3.18)
NAPVp^^^cCi)
Cq rfT'/d+Pf.)
NAPV^cCC-,) -
=
CE(C^)
TfT'
CQ(1+if./1-Tcc)
=
^0
^0 + ^0
i+r^.
I+Pf
1+rf
where the simple discounting rule has been used once more in
the calculation of NAPV^^(C^);
and T'
is still equal to T - 1/(1-1^^).
Equivalently equation (3.16) or (3.18) should be
solution
general
the
viewed
as
public project valuation proposed in the paper
to
.
This remains a fairly simple valuation rule.
In
sunmary the public valuation could involve twD steps.
in equation (3.16) the base case NPV can be calculated.
This involves
As
the
project direct cash-flows, the certainty equivalent operator and the social
riskless
rate
of
time preference.
Then in a second step is computed
distortions and displacements.
present
value
This first step is independent of tax
savings and investments displaced in the economy.
of
net present value (NPV^.) involves only the social
preference,
the
market
riskless
tax/displacement variable T.
equal
to
zero
the
rate
of
riskless
interest
and
rate
the
Because of the tax distortions NPV^
of
net
This
time
estimated
will
be
in the single case where only savings have been displaced.
Because the polar case where only investments are displaced is an important
benchmark since it
corresponds
to
the
optimal-tax/production-efficiency
35
assumption,
have
we
distinguishes
itself
displacements
between
proposed
by
displacements
in
the
splitting
first
a
efficiency assunption and
equivalent
the
a
rule
present
value
of
than
induced
which corresponds to the production
term
second term which corresponds
"excess" of the neo-classic assunption;
induced
the
to
i.e.
four terms of (3.18) would lead to accept or reject projects
basis
This rule
(3-18).
on
the first
the private sector, Ihe public valuation involves a fifth term
because it is assuned that the project might not have been financed
out
crowded out private investments.
of
same
the
solely
Notice that in the NAPV approach
adding
additional cash-flows externalities could easily be handled by
new
terms to (3-16) or (3-18).
certainty
As pointed out in the
calculation
remains
extremely difficult.
estimate
to
T
(or T').
case
the
hazardous
only
Unfortunately this must be
T depends on elasticities of savings and investments.
But in any case the adjusted present value approach does not complicate the
valuation beyond the steps that were required in the adjusted discount rate
approach.
The
result
demonstrated
for
the
valuation
of
the
alleged
displacements, should stress that the elasticities need to be estimated for
the
different
effective
classes
of
marginal
taxation, but need not be
related to the different classes of risk-adjusted returns.
A last warning about the cost of
(in
(w)
3-16)
displacements.
The
i^,7 is in no way equal to the weighted average cost of capital
proposed by Harberger and in practice, generally estimated and used
public
term
valuations.
in
"w" is valued at the marginal return on assets in the
economy and is inclusive of the risk premia!
Before reexamining Bailey and Jensen's solution in the light
of our results let us give the multiperiod (perpetuity) solution, when only
36
the first outlay generates
project
costly
displacements:
in
every
period
the
generates an uncertain cash- flow with constant expected value (C),
and the initial outlay is Cq^ then
^^P^b^C)
=
NPVb + PVf(Co)
=
C/r^ - Cq +
=
C/r^ + Cq
(
CqCI- T/(1-Ti))
T/(1-Ti)
)
37
SYSTEMATIC ERRORS IN BAILEY AND JENSEN'S APPROACH
IV.
Bailey and Jensen's Social Capital Asset Pricing Model (SCAPM):
Bailey
argued
Section III, that a reasonable guess might be to assune that the degree
with
optimality
for
They conclude as we did in
risk in public project valuations.
considering
correctly
have
Jensen[1972]
and
of
to risk sharing on public assets is comparable to
regard
the one achievable in a private stock market.
With this assunption in mind
we ought to use publicly traded securities to
extract
information
projects:
the
on
two
the
most
reliable
prices necessary for adequate valuation of risky
investors' riskless rate of time preference
and
the
price
of
risk.
In
of
Harberger's
government
other words Bailey and Jensen want to restate the problem
externalities
ownership
does
not
correct measure of the riskiness
contribution
following
distort
of
a
Assume
terms.
that
improve risk bearing, if the
or
public
project
is
its
marginal
to the standard deviation of the market portfolio (as a proxy
for individuals'
assumptions
the
in
of
true
the
portfolio),
Siarpe-Lintner
i.e.,
CAPM.
its
beta
Then
we
under
the
standard
can derive a set of
adjustments which account for the tax distortions in the public valuations.
This is exactly what were the assunptions of our derivation.
38
In order to solve this problem they
explanation,
propose,
without
much
the following Social Capital Asset Pricing Model, which, they
claim, provides the
appropriate
social
risk
(i.e.
preraiuns
including
externality adjustments) to apply to individual projects.
The market-determined equilibriun private expected rates
return can be described by the traditional CAPM relation
(^•1)
E(
ij^)
=
if
h
+
.
of
:
(private or
Beta,^
market rates)
where
the expected return on asset k (after
^^ijic) is
before personal taxes)
corporate
but
taxes
if is the riskless rate
h is the market price of risk per unit
^^^m) - if
i|vi
/
)
of
standard
deviation
=
(
derive
a
Gm
is the return on the market
C^ is the standard deviation of the market return
Betaj^ = Gov
ik
(
,
iM
)
/ Gn,
Bailey and Jensen claim that it
CAPM-like relation for social discount rates
(n.2)
E(
Wj^)
-
wf
+
k
.
Betaj^
is
possible
to
:
(Public Rates)
39
where,
(U.2.1)
w^
i^
=
E
[
(1-Tij).dSj/dB -
(4.2.2)
Wj^
if
=
i^
L
(l-Tip.dSj/dB -
(4.2.3)
k
ik
=
h
h
=
.
equation
E
J
"^
to
signs).
drive
1/(1-T^,,^).dVdB
]
1,
1
the
in
previous
the
section,
implies that each marginal personal tax rate
(2.0)
above
T
.d
T
will tend to drive T below
tend
]
E(l-T^j).dSj/dB -
Notice that as pointed out
displacement
VdB
l/d-T^i^)
T
.
[
E
k
J
=
]
T
.
[
1/(1-T^j^).dVdB
"^
J
=
E
and each marginal
corporate
tax
rate
will
long dSj/dB and dlj^/dB have the expected
(as
The net effect is uncertain:
>
T
=
1
<
It can go either way
borrowing
on
savings
depending
on
the
relative
effects
of
government
and investments along with the relative size of the
tax distortions.
No proof is provided, but at
surprising.
average
The
return;
elasticities
are
instead
constant
the
no
risk-free return and the risk premiums!
first
glance
the
result
is
longer used to compute a weighted
term
(T)
multiplies
both
the
40
The adjustment of the price of risk for tax distortions
hardly find a theoretical justification.
1
,
why
can
If it happens that T is less than
should the government apply to the valuation of the projects lower
risk premiums than those which private investors would apply?
On the other hand none of the
fran
the
equations
in
proceed
(4.2)
rationale as Harberger's weighted average w (in 2.2).
sane
new version suffers from clear inconsistancies
that
absent
were
in
The
the
riskless version^^.
The Bailey and Jensen formula does
decision
rule
for
the previous section.
social
not
provide
right
the
risk adjusted public valuations, as forraaly derived in
Ultimately we shall see that the error made
in
the
CAPM is very sijnilar to that in the Modigliani-Miller adjusted cost
of capital.
systematic errors
The costs of
of
outlay
the
government
the
alleged
project
is
displacements
very
similar
on
to
the
original
the adjustment
performed in private project valuations to account for interest tax-shields
on the project's financing.
We can now anticipate that Bailey
and
Jensen
did not actually derive a Social CAPM but rather performed an adjustment on
risky
discount rates very much like Modigliani and Miller Adjusted Cost of
Capital.
We can notice the similarity between the two formulations:
41
1.
Modigliani and Miller cost of Capital:
=
^c
^c
1-TccI^
(
)
where
i^ is the risk adjusted cost of capital on an all-equity firm
Tqc is (as before) the marginal corporate tax-rate
is the project marginal contribution to the firm's debt capacity
^
2.
Bailey and Jensen's social cost of capital
=
'^c
ic
T
•
As it is
decision
rules
finite
for
lived
adjustments
those
known
well
projects
decision rules become exact for perpetuities.
cost
adjusted
the
of
capital
will
Myers[1974])
but the
cases
exactly
equals
zero.
In
words, even in the perpetuity case the adjusted cost of capital just
since often
exclusive
the
government
projects
NPV.
This
worrisome for the kind of decisions to be undertaken
more
potentially
of
might
consider
resources
for
two
mutualy
very different kinds and for which we might expect
different induced displacements (and hence different T)
might
,
both
Nevertheless in
defines a hurdle rate, and does not necessarily give the right
is
inexact
to
lead to the wrong net present value
calculations except when the net present value
other
(see
lead
.
Both
valuations
display positive NPVs but the magnitudes of the calculated NPVs will
be wrong.
The
decision
rule
but
adjustment
still
could
of capital.
This
(providing
rescued
the wrong NPV calculations)
case if performed with riskless rate
valuations.
be
of
return
on
the
right
in the finite-lived
certainty
equivalent
also would apply to Modigliani and Miller adjusted cost
42
In order to see these
results,
let
consider
us
the
one
period project of Section III.
Bailey and Jensen propose the use of
w^
=
i^.T
=
(
:
Tc/Cl-Ti) ).T
Hence they compute the following net present value
UC
(4.3)
NPVb4j(Ci)
:
E(Ci)
)
Cq =
=
Cq
1+ic-T
l+'*c(1/''-Ti)-T
E(C^)
.
=
1+[i"f+
E(C^)
Co
h.Cov(rc.rM)](1/l-Ti).T
_
Co{l+[rfv^ h.Cov(rc.rM)](1/l-Ti).T}
1+[^fv^
h.Cov(rc.rM)](l/1-Ti).T
The APV calculations in chapter III lead to the result (3.16)
CE(C^)
^PVput,(Ci)
.
Cod.ifT)
=
1+r^
E(C^) . h.Cov(Ci,rM) - Co(UifT)
1+r^
E(C^) . h.Cov(Ci,rM) - Co( 1+rf( 1/1-1^)1)
1+r^
which can be rewritten
E(C^) _ h.Co.Cov(ipc,rM) - Co(
Urf( l/l-Ti)T)
(4.4)NPVp^^(Ci)
1+r^
where,
ipQ is defined as the project return (not
specific market return)
necessarily
equal
to
any
43
Clearly (4.3) and (4.4) will not lead in general to the same
decision rule, as long
^^^^.rM)
=
Cov(rc,rM).(1/1-Ti).T
No simple comparison between the
made
but
in
some
cases
two
calculations
can
be
Bailey and Jensen's approach will also give the
wrong sign.
We can propose an alternative rule to
adjusted
discount
rate
Bailey
and
Jensen's
by limiting the adjustment to the riskless rate
Wf-i^T, and using Wf in certainty equivalent valuations.
CE(C^)
1+w^
:
44
adjustment
nor need be, performed on the price of risk.
is,
more that w^ is not equal to SDR "w"
,
(Notice once
and current estimates of w are of
no
use for the proposed modified adjustment rule)
Finally, in a multiperiod framework (perpetuities, see
equation of chapter III)
^APVb(C)
:
C/r^ + Co(T/(1-Ti)
=
=
[C - CQi^T]/r^
but according to Bailey and Jensen's rule
NPVb4j(C)
=
C/i^T - Cq
In
present values.
last
=
:
[C - CoicT]/ieT
both cases Bailey and Jensen's rule provides inexact
net
45
V GENERALIZATIONS AND CONCLUSION
valuation
We have proposed in the paper a simple
rule
for
public investments.
The valuation rule can be sunmarized in the following way
:
the net present value of the project is equal to the base-case NPV plus the
present
of
value
displaced
savings
and investments, computed as if all
returns were equal to the riskless rate and discounted back at
discount
riskless
rate
proper
This reflects the application of Fischer Black's
rate.
simple discounting rule.
the
In this
definition the proper
riskless
discount
the social rate of time preference, which we have set to be equal
is
to the marginal rate of time preference in the private sector, although the
result
would
assunption.
carry
through
However,
the
under
public
reasonable
sector
must
departures
account
for
from
this
and
price
systematic risk in the same way as the private sector.
This simple valuation rule is appealing not only because
corrects
the
systematic
errors
made
discount rate approach or in Bailey and
it
in the traditional adjusted social
Jensen's
social
CAPM,
but
also
because it can be easily generalized to many valuation problems.
Valuation of projects cannot
discounted
cash- flow
rules.
always
be
dealt
valuing
of
these
This is often the problem confronted in public valuations since
many projects are energy related (oil projects are typically treated
series
using
When the underlying real asset is a sequence
of real options, only contingent claim analysis can help in
projects.
with
real
options
on
as
a
future production) or are linked to explicit
guarantees (the government commits itself to subsidies and bail
outs
that
46
only
in
certain
discount rates can
be
of
occur
would
no
of the world)
states
help
in
adjusting
.
Obviously adjusted
for
the
hypothesized
displacements, the valuation of such options.
rule
The simple valuation
relation
market
among
returns
is
holds
better
in
case
the
described
more
by
the
where
general
risk-return relationships, as in Merton's[ 19733 intertemporal asset pricing
model, or Ross'
be
shown
[
1975, 1976] arbitrage pricing theory.
that
the
In particular it
can
last proposition is very general and that the present
value of excess equilibrium returns is always equal to zero.
Also the paper along with the above discussion
to
practical
in practice.
project
with
and
lead
recommendations for the implementation of the valuation rule
The APV distinguishes between
the
valuation
of
the
basic
the proper project risk considerations, and the valuation of
the financing externality.
invested
should
is
latter
The
independent
of
is
proportional
to
the
amount
the risk characteristics of the project
(this is actually the main reason why Bailey and Jensen's rule is wrong)
The
approach could also draw the attention to those projects that are only
profitable because of favorable hypothesized shifts in savings.
B
TFB
1
FOOTNOTES
Lessard, Bollier, Eckaus, Khan [1982] point out that although currently
exists substantial controversy over the 'precise' estimation of the social
discount rate "w" (the results range from 7 to 10 percent or more in real
the general approach has wide acceptance in Canadian public policy
terms )
analyses.
'
,
2 Some SDR theorists acknowledge the absence
of uncertainty in their
approaches but either they argue that the problem would justify a separate
or
they take for granted Bailey and
paper (Sandmo and Dreze[ 1971 ])
Jensen's results (Sjaastad and Wisecarver[1977]) which as we shall see
seem to accredit a mostly unmodified transposition of SDR adjustments in a
multi-rate context.
,
,
Jenkins[1981] even argues in the case of the public valuation of
oil project against the use of the private opportunity cost of capital
computed in the oil industry because the low return sectors such as
"aircraft and parts, railways, water transport and grain elevator" brings
the social opportunity cost of capital down and hence makes it a more
reliable estimate of what would be the true opportunity cost of public
funds.
tax wedge here, only of discrepancies
No mention is made of
attributable to difference in risk preraia!
3 Glen
an
^ Other approaches than the one considered here
often state that the
might have no links
preference
discount
and
social
rate
of
time
social
rate
take
into account such
with market rates because cost-benefit analyses must
capital
poorly
developed
externalities as:
distribution externalities,
markets, absence of intergenerational bequest motives... On the other hand
Harberger states that his theory is adapted only to the case of well
developed economies with well functioning capital markets where these
externalities can be left aside in the analysis.
See Stiglitz[1982] p. 165 for a simple derivation.
Alternatively, if
the economy is originally in a state of production efficiency, the private
sector will not increase the level of saving when "offered" a public
project with a return between the before-tax and after-tax rate of
interest, and private sector investment will be crowded out more than one
Ihis is the
for one (I thank Fischer Black for pointing this out to me).
case because rational investors will anticipate that the loss of tax
revenues will induce the government to raise taxes on future income. This
increase in the tax rate must induce a decline in national wealth,
otherwise we could not have been at a point of production efficiency
other
In
initially.
This contradicts the hypothesized shift in savings.
words once the government has chosen the level of interest rate and has
designed its taxes in consequence, consunption levels are fixed, and a unit
of public investment must crowd out at the margin a unit of private
investment.
5
^ Once we relax the initial assunption of optimal
control, we cannot
obtain the strong result.
If the economy was not in a production-efficient
equilibriun in the first place and if the same experiment as in footnote 5
was performed, they might not even be a need for a shift in the tax rate
for
the government to protect its future tax revenues.
There will also be
TFB
some uncertainty with regard to the behavior of the government. One's best
guess might then be that the tax rates are fixed at the current level for
all the relevant horizon.
Under these circunstances public investment need
not crowd out one for one private investments and equation (2.0) does offer
a more general approach to the impact of public investment.
does not mean that all kind of subsidized investment is banned
such economies but rather that only an equilibriun is reached where
none of the private sector with higher return than the project to be valued
can be efficiently subsidized or publicly owned.
7 This
from
° Notice that this would not be a constraint
under the conditions of
Diamond and Mir rlees[ 1971] since a reduction in the rate of taxation is
another way for the government to induce the undertaking of the marginal
private project. The government could adjust the tax rate until production
efficiency is achieved.
9 Recently Arrow[1982], developing Arrow and Kurtz[1970], has looked
at
raultiperiod growth models.
They show that the opportunity cost of capital
remains in these contexts a weighted average of consunption rate of time
preference and marginal before-tax-return on private capital, but the
weights would be very difficult to observe, project specific (related to
the complementarity of private and public production) and related to other
economic variables.
However Sjaastad and Wisecarver[ 1977] argue that
reinvestment of the net project output has negligible quantitative effects.
They also propose to treat as "external" additional benefits to be
attributed to the project, with the augmented benefit stream then being
discounted by "w"
Lind[1982] makes similar advice. This would also be
very well adapted to our adjusted present value rule in the case of risky
.
environments.
^^ An
alternative version of w is also
frequently used in SDR theories
(2.1)'
w =
- g dl/di)
r dS/di
(
/
or in tenns of interest elasticities
given
by
Harberger
and
more
:
(
dS/di - dl/di
)
:
(2.1)"
w=
(rSe3_ginj-)/(Ses-Ini)
those expressions are equivalent. The 'borrowing' version is simpler
since if all markets are in equilibrium, dS/dB - dl/dB =1.
It is also the
one used by Harberger when he discussed the possibility of different yields
due to risk premia, and seems more intuitive in that context.
All
^^
In the small
economy case, Harberger[ 1976a, 1976b]
(or
see Hood,
Glenday,
Evans[1982])
introduces country risk in his model: an upward
slopping cost of foreign funds drives the increase in the interest rate
paid on savings and wnich feeds back in the crowding out of investments.
'"^We can notice that the neo-classical rule assunes more than that
only
private investments are crowded out, but makes additional assunptions on
the homogeneity in terms of risk and taxation of the private projects
displaced. Hence the displacement approach might be formulated even in the
case where savings are not created in the process, but where we need to
take into account the heterogeneity of taxation.
13 I
thank Stewart Myers for suggesting this formulation.
TFB
1^ This is a stronger condition than we actually need for a market
CAPM
to hold (on before personal tax returns).
As discussed in Long[1977], what
he calls the "efficiency equivalence hypothesis" can be met under fairly
general description of the tax structure.
B
TFB
1
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,
,
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