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A review of the academic literature concerning the determination of the social rate of discount

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A review of the academic literature concerning the determination of the social rate of discount
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Scott, Glenn Eugene
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Submitted in partial fulfillment of the requirements for the degree, Master of Arts, Economics.
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Department of Economics
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by Glenn Eugene Scott.

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Full Text
A REVIEW OF THE ACADEMIC LITERATURE CONCERNING
THE DETERMINATION OF THE SOCIAL RATE OF DISCOUNT
by
Glenn Eugene Scott
B.A., University of Colorado, 1985
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Arts
Economics
1992


This thesis for the Master of Arts
degree by
Glenn Eugene Scott
has been approved for the
Department of
Economics
by
W. James Smith
Steven Medema
Suzanne W. Helbum
Date


Scott, Glenn Eugene (M.A., Economics)
A Review of the Academic Literature Concerning the Determination of the Social
Rate of Discount
Thesis directed by Professor W. James Smith
ABSTRACT
Irving Fisher demonstrated that the social rate of time preference and the
social opportunity cost of capital are equalized at the market rate of interest under
first-best conditions. The result of relaxing the first-best assumption is any number
of social rates of discount reflecting the particular conditions which are allowed.
The conditions which distort the market rate of interest as a measure of the
social rate of discount are addressed by two schools in the literature. One school,
led by Amartya Sen and Stephen Marglin, advocates the use of the social rate of time
preference as the social rate of discount. Another school, led by William Baumol,
advocates using measures of the social opportunity cost of capital.
The conditions affecting the social opportunity cost of capital are often project
specific, so that there is no universal social discount rate. Rather, the rate
appropriate to each project is dependent upon the nature of the constraints upon first-
best optimality resulting from the circumstances of that particular project.
A third school begun by Otto Eckstein and developed by Partha Dasgupta,


David Bradford, Robert Lind, and Joseph Stiglitz advocates discounting at the social
rate of time preference to the extent it is imparted by the rate of interest paid by
consumers. Opportunity costs of capital diverted to the public project are taken into
account by employing a shadow price of capital.
This abstract accurately represents the content of the candidates thesis. I recommend
its publication.
Signed
W. James Smith


CHAPTER
CONTENTS
INTRODUCTION 1
Importantance of the Social Rate of Discount 3
Overview of Thesis 4
Thesis Statement 6
1. THE FISHERIAN DECISION CRITERION 7
The Discounted Present Values of Net Benefit Streams 7
Decision Criteria 9
The Relationship between Discounting and Decision Criteria 10
Fishers Marginal Rate of Return over Cost 12
The Investment Opportunity Curve 15
Conceptualization of Time-Preference 17
The Rate of Interest at Equalibrium: Fishers Theory of Interest 19
Incorporation of Uncertainty: Fishers Third Approximation 20
The Keynesian Marginal Efficiency of Capital 22
Fisher, Keynes, and Decision Criteria 24
Conclusion 30
2. DISTORTIONS TO THE MEASUREMENT OF THE
SOCIAL RATE OF TIME PREFERENCE 34
Introduction 34
The Authoritarian Solution 36
The Schizophrenic Solution 38
The Interdependent Solution 39
Conclusion 48
3. DISTORTIONS TO MEASURES OF THE MARGINAL RATE OF
RETURN ON INVESTMENT AND THE SOCIAL OPPORTUNITY
COST OF CAPITAL 52
Introduction 52
The Interest Rate on Government Bonds 52
The Rate of Return on Private Investment 54
The Social Opportunity Cost of Capital 56
4. THE SHADOW PRICE OF CAPITAL APPROACH 65
Dasgupta et al.s Decision Criterion Incorporating the
Shadow Price of Capital 66
Bradfords Shadow Price of Capital Decision Criterion 69


Conclusion
72
BIBLIOGRAPHY
73


Nomenclature
The market rate of interest
The social opportunity cost of capital
The number of time periods in a benefit or cost stream
The number of individuals in a generation
The rate of discount applied to a benefit or cost stream
The marginal one-period rate of return on private sector capital
The social rate of discount applied to a benefit or cost stream
The social rate of time preference
The internal rate of return
The rate of taxation
The corporate sectors portion of GNP
The proportion of output devoted to private investment (Dasgupta)
The portion of corporate financing achieved through equity
The rate of return in the private sector
The social rate of return in the private sector
The consumption of the ith individual
The consumption of the future generation
The consumption of the individuals own generation.
The interest rate on consumption
The marginal utility the individual receives from consumption in the


next generation
0
k
The marginal utility the individual receives from consumption by
members of his own generation other than himself
5
The marginal rate of transformation
The social value of a unit of private capital in some time period
measured in terms of consumption, a.k.a the shadow price of capital


INTRODUCTION
My initial intent for the thesis was to conduct a cost/benefit analysis of
workfare programs both from the recipients and societys points of view. As I
explored various techniques of evaluation, I found I could not conduct any sort of
objective evaluation without assuming a rate of discount. However, I had no basis
for selecting a rate of discount which was not for the most part arbitrary. My search
of the literature provided no easy answers. I became troubled by the irrelevance of
objective public policy evaluation methods given their use of subjectively determined
rates of discount. It occurred to me that, where subjectively determined rates were
used, the analysis of public projects became merely a partisan exercise where a
politically chosen discount rates determined the outcome.
I also became more aware of the need for precision in selecting the correct
rate of discount given the sensitivity of present values to small changes in the rate of
discount. Table 1 illustrates this sensitivity. A twenty year net benefit stream is
discounted at ten different rates. In year zero, a $100,000 outlay produces a net
benefit stream of $16,301 per year for nineteen years followed by a shut down cost
of $250,000 in the last year. The effect of changing the rate of discount from three
to four percent results in a change in the net discounted present value from -$4,926
to $0.
1


TABLE 1 NET BENEFIT STREAMS
Discounted Discounted Discounted Discounted Discounted Discounted Discounted Discounted Discounted Discounted
Year at0% at 1% at 2% at 3% at 4% at 5% at 7% at 9.6367% at 12% at 15%
1992 (100,000) (100,000) (100,000) (100,000) (100,000) (100,000) (100,000) (100,000) (100,000) (100,000'
1993 16,301 16,140 15,981 15,826 15,674 15,525 15,235 14,868 14,554 14,175
1994 16,301 15,980 15,668 15,365 15,071 14,786 14,238 13,561 12,995 12,326
1995 16,301 15,822 15,361 14,918 14,492 14,081 13,306 12,369 11,603 10,718
1996 16,301 15,665 15,060 14,483 13,934 13,411 12,436 11,282 10,360 9,320
1997 16,301 15,510 14,764 14,061 13,398 12,772 11,622 10,290 9,250 8,104
1998 16,301 15,356 14,475 13,652 12,883 12,164 10,862 9,386 8,259 7,047
1999 16,301 15,204 14,191 13,254 12,387 11,585 10,151 8,561 7,374 6,128
2000 16,301 15,054 13,913 12,868 11,911 11,033 9,487 7,808 6,584 5,329
2001 16,301 14,905 13,640 12,493 11,453 10,508 8,867 7,122 5,878 4,634
2002 16,301 14,757 13,373 12,129 11,012 10,007 8,287 6,496 5,248 4,029
2003 16,301 14,611 13,110 11,776 10,589 9,531 7,745 5,925 4,686 3,504
2004 16,301 14,466 12,853 11,433 10,182 9,077 7,238 5,404 4,184 3,047
2005 16,301 14,323 12,601 11,100 9,790 8,645 6,764 4,929 3,736 2,649
2006 16,301 14,181 12,354 10,777 9,413 8,233 6,322 4,496 3,336 2,304
2007 16,301 14,041 12,112 10,463 9,051 7,841 5,908 4,101 2,978 2,003
2008 16,301 13,902 11,874 10,158 8,703 7,468 5,522 3,740 2,659 1,742
2009 16,301 13,764 11,642 9,862 8,369 7,112 5,160 3,412 2,374 1,515
2010 16,301 13,628 11,413 9,575 8,047 6,773 4,823 3,112 2,120 1,317
2011 16,301 13,493 11,190 9,296 7,737 6,451 4,507 2,838 1,893 1,145
2012 (250,0001 (204,886) (168,243) (138,419) (114,097) (94,222) (64,605) (39,703) (25,917) (15,275"!
Net Discounted Present Value (40,280) (24,084) (12,668) (4,926) 0 2,781 3,876 0 (5,847) (14,238:
2


Importance of the Social Rate of Discount
There exist considerable differences of opinion regarding the determination
of the social rate of discount and corresponding decision criterion, despite a strong
consensus on their importance. Eckstein [1957, p. 56] for example devoted himself
to developing "the kind of criteria that are needed for project selection." In a
subsequent article, Eckstein [1958, p. 94] states, "the choice of interest rate for the
design and evaluation of public projects is perhaps the most difficult economic
problem and yet one of the most important ones faced in this field." Feldstein
[1964a, p. 117] states that "economics must provide criteria for evaluating the
desirability of undertaking particular projects and for choosing between competing
public investment options." Baumol [1968, p. 788] warns of "serious misallocation
of resources [which] can result from an incorrect estimate of the [social rate of
discount] in cost-benefit analysis."
Despite these calls we find Lind et al. [1982, p. 18] years later urging then-
colleagues to addresses "the unresolved issues pertaining to the choice of the discount
rate to be used in benefit-cost evaluation of public projects." Lind [1982, p. 88]
states further, "there must be clear instructions to the analysts about what discount
rate to use or how to compute the appropriate rate of discount for different projects.
... The analyst simply cannot have discretion in the choice of the discount rate." It
has been over thirty years since Ecksteins comments, and hundreds of pages of
3


relevant material have appeared in the major economics journals; yet, Linds 1982
appeal bears great similarity to Ecksteins.
There are several contexts in which the social discount rate is applied. One
context common in the literature is the cost-benefit analysis of large, publicly funded
infrastructure projects (particularly related to water and energy). Also, much of the
literature concerning the social rate of discount is in the context of the socially
optimal level of investment. Recently, the social rate of discount has become
important for its use in the analysis of the "social investment" of public pension
funds. Our society expends considerable resources to measure the costs and benefits
of various public activities. It is a tremendous waste when the bottom line is varied
by large magnitudes due to subjective alteration in the rate at which we discount.
Overview of Thesis
This thesis asks the following central question: What does economics have
to offer those analysts who require a rate at which to discount when conducting an
evaluation of public projects?
In Chapter 1,1 examine and define the terminology of the literature, look at
the present value and internal rate of return methodologies, and outline Fishers
determination of interest rates under equilibrium conditions.
Fisher [1930] also introduced the distortionary effect of risk upon the market
4


rate of interest as the social rate of discount. In addition to risk, there are numerous
conditions which create distortions between the social rate of discount and the market
rate of interest. These conditions fall into two types. One type distorts
measurements of the opportunity cost of capital, the other effects the measurement
of the social rate of time preference. In Chapter 2,1 examine the conditions which
affect the determination of the social rate of time preference. I focus on the writings
of Sen and Marglin. Their writings center on conditions which prevent markets from
reflecting the social rate of time preference.
In Chapter 3, I examine conditions which distort the reflection of the social
opportunity cost of capital in markets. In particular, I discuss Baumols 1968
benchmark work which traces the effects of the corporate income tax and risk upon
the determination of social opportunity cost of capital.
Chapter 4 examines the shadow price of capital method for discounting public
investment from the point of view of society. Simply put, the method is to discount
at the social rate of time preference and then adjust by the shadow price of capital.
Since the shadow price of capital will vary from project to project, no two projects
will be discounted and adjusted in exactly the same way. I conclude with a
procedural outline and decision criteria for the public project evaluator.
5


Thesis Statement
This thesis is a comprehensive review of the academic literature concerned
with the complications of discounting in the evaluation of public investment. It
explores what economics offers to the analyst of public investment in need of a social
rate of discount.
Thesis Statement: Fisher demonstrates that the market rate of interest
equalizes the social time preference with the social opportunity cost of capital
under first-best conditions. The result of relaxing the first-best assumption is
numerous social rates of discount reflecting the particular conditions considered.
These conditions are often project specific so that there is no universal social
discount rate. Rather, the rate appropriate to each project is dependent upon
the nature of the constraints upon first-best optimality resulting from the
circumstances of that particular project. The public policy analyst, conducting
a cost-benefit type ranking or screening of public projects, needs to consider a
number of characteristics specific to the project. This is best achieved by
measuring the shadow price of capital for each project and discounting at the
social rate of time preference.
6


CHAPTER 1
THE FISHERIAN DECISION CRITERIA
The Discounted Present Values of Net Benefit Streams
A net benefit stream is composed of benefits net of costs accruing across
distinct time periods. The immediate time period is labeled zero and is followed by
all subsequent periods through time period n. The value of n may be either a discrete
time horizon or the point in time where net benefits resulting from the particular
activity cease. In the latter case, the definition of cessation becomes problematic if
net benefits are permanent and no weight or preference is given to net benefits
accruing sooner rather than later. In practice, the problem is overcome by
discounting, which is the process of assigning a weight or preference to a net benefit
which occurs sooner rather than later. This process causes the present value of net
benefits to diminish as the projects life increases.1
1 However, there are economists who advocate no time preference. Ramsey
[1928, p. 543] argues that the appropriate rate of discounting future utility to the
present is zero and ridicules those who possess a "fundamental weakness of the
imagination" for wrongly placing the present ahead of the future. In order to
overcome the complications of a zero rate of discount, Ramsey postulates the
existence of a negative marginal productivity of capital over and above a saturating
abundant stock of capital. Eckstein [1958, p. 75] criticizes Ramseys rejection of
discounting future utility by arguing for respect of consumer sovereignty, "including
their intertemporal preferences."
7


The introduction of discounting explicitly affects the choices of time horizons,
for as the rate of discount increases the relative magnitude of more futuristic net
benefits decreases (ultimately to insignificance). The effect of discounting on relative
magnitudes is profound. For example, a $100,000 benefit accruing in one hundred
years discounted at 10% is worth $7 at present.
The net discounted present value, net discounted present value, of an
investment stream is the sum of all the net benefits when discounted to their present
value. Algebraically, given a stream of net benefits,
where Bt is any number, the present value may be expressed as,
B0+BIl(Ur?+BJ(Urf+...+Bja+rr
which reduces to,
Eiya+ry
t=0
where r is rate of discount.
In this thesis, I am concerned with the point of view of society. It is assumed
that this point of view and the associated values of the social welfare function are
expressed through democratic processes. The rate of discount, r, appropriate for this
point of view is the social rate of discount, d. At equilibrium under assumptions of
8


perfect capital markets, d is measured by the market rate of interest, i. However,
t
when we begin to allow real world distortions, then d ^ i, and d may be best
measured by the social rate of time preference, p, the social opportunity cost of
capital, 7, or some other measure of the time-value relationship. Before considering
these distortions, let us consider the argument for d=i and the decision criteria which
result.
Decision Criteria
Cost-benefit analysis incorporates net discounted present values into decision
criteria. A common decision criterion is the discounted benefit/cost ratio greater than
one. A discounted benefit/cost ratio is the sum of the present values of all benefits
divided by the sum of the present value of all costs. Refer again to Table 1 and
specifically the second column (where the discount rate is 1%). We have an initial
outlay of $100,000 in 1992 and $75,916 in discounted benefits accruing from 1993
through 2012. The benefit/cost ratio is equal to $75,916/$100,000 or .75916. Since
this ratio is less than one, we would reject this hypothetical project. This is an
example of a screening criterion; that is, a project is either accepted or rejected
relative to a standard.
Screening criteria are different from ranking criteria. Under a ranking
criteria, projects are ranked according to some measure of their net benefit, for
9


example their benefit/cost ratios, from highest to lowest. Projects are then selected
until either available resources are exhausted or some other constraint is reached.
There are numerous other decision criteria all of which share the common element
of a rate of discount applied to their net benefit streams.
The Relationship between Discounting and Decision Criteria
Marglin [1963b, p. 277] provides an illustration similar to Graph 1 to clarify
the necessity of discounting. In Figure 1.1, two projects, A and B, and their
corresponding net benefit streams are illustrated. Regardless of the discount rate
chosen, project B will always have a greater net discounted present value and
benefit/cost ratio than project A. Further, in all ranking type decision criteria,
project B will always be ranked ahead of project A. However, under a screening
criteria, we are not provided with enough information to say whether projects A
and/or B will be accepted or rejected. We do know that if a particular screening
criterion rejects project B it also must reject project A.
In Figure 1.2, the lines intersect. This situation demonstrates that Project A
has a lower rate of return initially but it eventually exceeds Project B. Which project
should the public sector prefer? Without a social discount rate the answer is
ambiguous. An ad hoc or arbitrary rate will give an arbitrary result. The
relationship between the two projects benefit streams is such that the selection of a
10


FIGURE
1.1
FIGURE 1.2
PROJECT A PROJECT B
11


relatively low rate of discount will result in the discounted benefit/cost ratio of
project A exceeding that of project B. Likewise, at higher rates of discount Project
Bs ratio will be greater than project As.
To avoid arbitrary results we need a social rate of discount derived rationally
to apply to both ranking and screening decision criteria. The need for a correct
social rate of discount remains constant and essential, regardless of whether we
choose a screening or ranking type of decision criterion. There are two schools of
thought surrounding the discounting of net benefit streams, Fishers marginal rate of
return over cost and Keynes marginal efficiency of capital. The search for a correct
social rate of discount begins with a choice between these two schools.
Fishers Marginal Rate of Return over Cost
The decision criterion advocated by Fisher is to select that particular bundle
of projects the comparison of which to any other bundle results in a rate of return
over cost equal to or in excess of the market rate of interest. Project intensities are
ranked according to their marginal rate of return over cost and accepted until the
constraint of the market rate of interest is reached. There are two parts of Fishers
criterion which I will examine. The first part is the meaning and determination of
the rate of return over cost, and the second is the meaning and determination of the
market rate of interest.
12


Fisher [1930, p. 151] defines an investment option as "any possible income
stream open to an individual by utilizing his resources, capital, labor, land, money,
to produce or secure said income stream." An option may consist of one or more
particular uses of these factors. That is, there may be more than one use in any one
option. Eligible options are options for which some rate of discount exists that would
cause its present value to exceed the present value of all other options (assuming the
cost for each of the options is the foregone opportunity of the next best option).
Given any two eligible options, there exists some rate of discount which will cause
the present values of the two to be equal. This rate of discount is Fishers rate of
return over cost, where returns are all of the comparative advantages of the selected
eligible option over the foregone eligible option and costs are all of the comparative
disadvantages of the selected over the foregone.
Fisher [1930, p. 151] defines an investment opportunity as the "opportunity
to shift from one (option) to another." The decision of when to accept or reject an
investment opportunity is given by Fisher [1930, p. 152] in four equivalent
investment opportunity principles. These are:
1) The Principle of Maximum Present Value
2) The Principle of Comparative Advantage
3) The Principle of Return over Cost
4) The Principle of the Marginal Rate of Return over Cost.
The Maximum Present Value Principle states that the option with the greatest
13


present value when both its costs and benefits are discounted at the market rate of
interest should be selected over all other options.
The Comparative Advantage Principle begins with the comparison of any two
options. The option that should be selected will result in the benefits outweighing the
costs, when compared with any other option, when both are discounted by the market
rate of interest. Costs (disadvantages) are the opportunity costs of the foregone
option. In cases where the benefits precede the costs, Fisher shows that symmetry
allows us to view the comparison from the perspective of either option so as to
always have costs preceding the benefits.
The Return over Cost Principle is similar to the Comparative Advantage
Principle, in that the selected option is compared to any other foregone option. Here,
the benefits and costs are discounted by the market rate of interest and expressed as
a rate of return. The selected option will have a discounted rate of return over cost
equal to or greater than the market rate of interest.
The fourth Principle adds the assumption of infinite investment opportunities,
where a "continuous gradation" replaces a finite set of investment opportunities.
Marginal analysis is added to the Return over Cost Principle resulting in the Marginal
Rate of Return over Cost Principle which states that the selected option will have a
marginal rate of return over cost just equal to the market rate of interest. Continuous
gradation implies that options are bundles of individual projects and/or marginal
14


intensities of projects which are ranked from the greatest discounted marginal rate of
return over cost to the least. Projects and their intensities are selected until the
market rate of interest is reached. Projects are mutually exclusive only to the extent
that the resources available are limited.
The Investment Opportunity Curve
Graphically, the universe of available eligible options is represented by an
investment opportunity curve (Fisher calls it a line). In Figure 1.3, we begin with
some initial amount of resources represented by point A. The' individual faces the
investment opportunity to forego the option of point A for any point on or inside the
investment opportunity curve.
FIGURE 1.3
MARKET RATE OF INTEREST
15


Points on the curve, represent the maximum possible future income for every level
of present income. The curve also represents the continuous gradation of investment
opportunities available, that is, the set of eligible options. A movement from point
A to point G would mean the selection of an ineligible option an option for which
no rate of discount exists that would maximize its present value relative to all other
options. Point G is not an.achievable coordinate of present and future income given
the available resources and investment options.
The slope of a production possibilities curve is equal to the marginal rate of
transformation. The investment opportunity curve is a special application of the
production possibilities curve where the marginal rate of transformation is the
marginal opportunity cost of capital in terms of present and future income. The
curves convexity results from the law of diminishing returns and differing facot
intensities in the production of goods. So that, as we move along the curve from
point A, we rationally select the options that give us the most in return per unit of
foregone present income before we choose options that give us less.
Consider the relationship between points F and C. Both of these options give
us the same level of future income, I. Yet C requires the forgoing of less present
income than F. There is no allowable point that will give us I while allowing any
more present income than C. Clearly, we would prefer C to F. But, the
formalization of preference and indifference requires a second concept.
16


Conceptualization of Time-Preference
One explanation of the tendency for individuals to value the present over the
future is myopia. Myopia is the irrational or shortsighted preference given by
individuals to present over future income. Myopia, according to Ramsey [1928, p.
543] resultes from individuals "fundamental weakness of the imagination." Bohm-
Bawerk also included individuals weakness of will and uncertainty about the length
of their own life spans in his explanation of the preference of the present and interest.
However, Bohm-Bawerk did not limit his explanation of the preference of
present over future income to myopia. Conard [1959, p. 45] states that Bohm-
Bawerk used the term agio to encompass any "premium which people are willing to
pay for present income over a claim to future income." Agio, according to Bohm-
Bawerk, results from the relative scarcity of means to fulfill present needs versus
means to meet future needs, shortsightedness, and/or the technical superiority of
future over present capital goods.
Fisher based his time-preference (a.k.a. impatience) on the diminishing
marginal utility of income. For Fisher [1930 p. 65], time preference is the
"preference for a dollars worth of early real income over a dollars worth of
deferred real income."
Conard discusses two definitions of time preference which he finds in the
literature. In the first, time preference is defined as the individual preference of a
17


redistribution of a given income stream over time under a zero real rate of interest.
A positive real rate of interest may exist without this time preference. The second
is Fishers time preference concept. Fisherian time preference exists when an
individual prefers a redistribution of a given income stream over time under any
condition. Fishers time preference is a necessary condition for the existence of a
real positive rate of interest. In either case, myopia is not a necessary condition for
the existence of positive time preference.
In the example of points C and F in Figure 1.3, point C is preferred to point
F, because the bundle of income represented by point C includes all the income of
point F plus the additional present income represented by (P" P). The relationship
between points C and D is less clear. By the sacrifice of ten units of present income,
thirty-two units of future income is gained. As illustrated in Figure 1.4, there are
three relationships possible between points C and D. Indifference Curve 1 illustrates
the case where point C is preferred over point D. The preference of point D over
point C is illustrated by Indifference Curve 2. Indifference in the selection of either
point is represented by Indifference Curve 3 (note that these indifference curves are
mutually exclusive). The slope of an indifference curve is the marginal rate of
substitution. In the Fisherian context, the social indifference curves depict social
welfare equivalent distributions of present and future income. In this context, the
marginal rate of substitution is the social rate of time preference between present and
18


future income.
The Rate of Interest at Equilibrium: Fishers Theory of Interest
Given the constraint of the existing investment opportunity curve, society
should seek the highest social indifference curve. The fulfillment of this constrained
maximization is, in the general case, the determination of the market rate of interest.
At the market rate of interest, the marginal rates of substitution and transformation
are equal. The social discount rate, d, will equal the market rate of interest, i, the
FIGURE 1.4
INDIFFERENCE CURVE 3
19


social opportunity cost of capital, 7, the social rate of time preference, p, and the
marginal rate of return of capital over cost of the selected eligible option. Of the
infinite set of indifference curves, one curve will have a single point (or segment) of
tangency with all other nontangent points outside the investment opportunity curve.
The slope of the line through this point is equal to i and d. Figure 1.5 depicts this
static equilibrium.
Incorporation of Uncertainty: Fishers Third Approximation
Fishers third approximation was to generalize his model by incorporating
uncertainty. The third approximation demonstrates Fishers recognition of the
limitations of i at equilibrium. Conard [1959, p. 67] summarizes this recognition,
stating, "Fisher concludes that under conditions of uncertainty those equalities which
are assumed to exist under the first two approximations (interest rate, marginal rate
of return over cost, and marginal rate of time preference) become only idealized
tendencies toward equality. "2
Conard [1959, pp. 65-67] examines five distortions Fisher creates through the
2 Other economists subsequent to Fisher, argue that taxes, market imperfections,
and externalities (among other things) distort Fishers equalities.
20


FIGURE 1.5
- INVESTMENT OPPORTUNITY CURVE # IN Dl I 'I 'liRENCE CURVE ^ MARKET RATE OF INTEREST


incorporation of uncertainty.3 Three distortions affect the marginal rate of return
over cost and two affect the social rate of time preference. First, uncertainty causes
the realized rate of return to diverge from the interest rate. Second, uncertainty
requires that the choice of options take account of the varying degrees of risk
involved. Third, uncertainty, by limiting borrowing, may eliminate desirable options.
Fourth, because of uncertainty, the individual may not actually maximize satisfaction
because of misjudgment about what will bring satisfaction. Fifth, with uncertainty,
the market may not permit the individual to maximize even the individuals
anticipated satisfaction because the risk of lending to the individual may exclude the
individual from the loan market.
The Keynesian Marginal Efficiency of Capital
A method related to discounting is Keynes marginal efficiency of capital.
The marginal efficiency of capital, MEC, is also known as the internal rate of return,
X. This rate equilibrates the present value of the benefit stream with the present
value of the cost stream.4 Equivalently, the marginal efficiency of capital is the rate
3 These five impacts begin a body of literature examining the determination of
the social rate of discount under conditions of market distortion. The examination
of this literature constitutes the remainder of this thesis.
4 There is considerable confusion over the difference between Keynes MEC and
Fishers MRRC. The confusion begins by not recognizing that Fishers MRRC is the
22


of discount which makes the present value of the entire stream of benefits net of costs
exactly equal to zero. Algebraically, given a net benefit stream,
B0,B iyBv... Jin,where .B zOJor. all.i=0,\2.,...,n,
the internal rate of return is the quantity for which
Bn
B,
B-
B.

(1+X) (1+X)* 1 (1+X)2 (1+A,)"
-=0.
This may be expressed in reduced form as,
n
B.

=0
The internal rate of return principle leads to Keynes decision rule, that is, accept the
investment if,
rate at which the sum of the marginal benefit of one eligible option and the marginal
opportunity cost of a foregone eligible option is zero. If this marginal rate exceeds
i for all other foregone eligible options, then this project passes Fishers MRRC
criterion. Keynes MEC is the rate of discount at which the sum of the benefit
stream and the cost stream of a particular project is zero. If this rate exceeds the
opportunity cost of capital then the project passes Keynes criterion. Do not confuse
Keynes criterion with Keynesian criterion based upon the internal rate of return.
Keynes himself thought his MEC and Fishers MRRC were equivalent. Alchian
[1955 p. 938] was the first in the literature to point out Keynes error. Alchian
states, "(Keynes] erroneously alleged that his marginal efficiency of capital is
identical with Fishers marginal rate of return over cost." See Fisher [1930, pp. 151-
169] and Keynes [1936, pp. 140-141].
23


*>Y,
i.e. accept the project wherein the marginal efficiency of capital exceeds the private
sector opportunity cost of capital, y.
The Keynes decision rule is a screening-type decision criterion
requiring an exogenously determined measure of the private sector opportunity cost
of capital. In Table 1, the internal rate of return, X, was 1.58%. In this case,
assuming the opportunity cost of capital is 5%, the project would fail Keynesian
decision rule. Keynes decision rule dictates that we reject projects where the
internal rate of return is less than the opportunity cost of capital, that is where X >
5%. There are other rules based on the internal rate of return, for example the
pragmatic Keynesian decision criterion. Under this rule, an ordinal ranking of
projects by their internal rates of return is created, projects are accepted until funding
is depleted.
Fisher. Kevnes. and the Decision Criterion
Both screening and ranking criteria involve some form of discounting. The
basis of that discounting may be the market rate of interest or the internal rate of
return. Therefore, decision criteria may be placed into one of four categories as
illustrated by the following matrix.
24


Matrix 1 d=i X
Screening 1 2
Ranking 3 4
EXAMPLES
1) Accept projects in which the discounted benefit/cost ratio is greater than one
or disqualify all projects in which the discounted benefit/cost ratio is less than
one. Equivalently, accept projects where the net discounted present value is
positive (the net discounted present value criterion).
Accept that bundle of project intensities whose marginal rate of return over
cost equals the market rate of interest (Fishers marginal rate of return over
cost criterion).
2) Accept projects where X equals or exceeds the marginal opportunity cost of
capital (Keynes marginal efficiency of capital criterion).
3) Rank projects by their benefit/cost ratios discounted at d. Fund until available
resources are exhausted (the pragmatic benefit/cost ratio criterion).
4) Rank projects by X. Fund until available resources are exhausted (the
pragmatic Keynesian internal rate of return criterion).
The net discounted present value criterion is a screening type adaptation of
Fishers Maximum Present Value and Comparative Advantage Principles. Fishers
marginal rate of return over cost criterion is equivalent to the net discounted present
value except that it adds the assumption of continuous gradation.5
5The relationship amongst Fishers four equivalent principals is discussed in
Fisher [1930, p. 151-154].
25


Figure 1.6 uses an illustration similar to Baumols [1977, pp. 607-609], to
display the marginal efficiency of capital criterion. The marginal efficiency of capital
criterion concludes that this project be accepted because X > i, assuming i is equal
to the marginal opportunity cost of capital. The net discounted present value criterion
also requires that this project be accepted because the net present value discounted
at rate i is positive ($50,000).
Baumol discusses two disadvantages of the net discounted present value
criterion with respect to the marginal rate of return over cost. First, Baumol [1977,
p. 606], states that "net discounted present-value need not always have a negative
slope." Figure 1.7 illustrates this observation and its implications. The bell-shaped
function between net discounted present values and the rate of discount in Figure 1.7
is typical of projects with life-cycles characterized by three (-,+,-) stages. That is,
initially we have an outlay stage, followed by a stage of positive net benefits,
followed by a negative shutdown/clean up stage. The bell-shaped function results
because lower rates of discount place relatively greater weight to stage 3s negative
net discounted present value. Likewise, higher discount rates place greater weight
on stage 1. Table 1 shows numerically a case resulting in two internal rates of return
illustrated in Figure 1.7.
26


FIGURE 1.6
ffl
27


Shown here is a project with an initial outlay of $100,000, followed by 19
years of $16,301 net benefits, and ending with a $250,000 shutdown cost. Assume
that the market rate of interest is 5%. In this instance, the marginal efficiency of
capital results in ambiguity since X = 4% < i which implies rejection while X =
9.6367% > i requires acceptance. By contrast, the discounted present value
criterion requires acceptance since, at discount rate i = 5%, the net discounted
present value, $2,781, is greater than 0.
Suppose the opportunity cost of capital is 5 %, should we accept or reject this
project? Given that both a X of 4% and 9.6367% results in net benefits equal to
zero, the answer is ambiguous. The pragmatic Keynesian criterion, which does not
concern itself with achieving the socially optimal level of investment, is also
ambiguous. Do we rank this project at 4% or 9.6367%?
The second disadvantage of marginal efficiency of capital illustrated by
Baumol [1977, pp. 608-610] may arise in situations involving mutually exclusive
alternatives. Mutually exclusiveness is a common situation. For example, the
damming of a canyon to create either a recreational lake or a nonrecreational
reservoir. If we choose one, we can not have the other. Figure 1.8 illustrates this
condition. In this graph, the net discounted present value criterion offers clearer
guidance. If we, for simplicity, assume the cost of the two projects to be equal, then
at discount rate i, the reservoir is preferred to the recreational lake, since the net
28


Thousands
1U
IE 1.8
29


discounted present value of the reservoir (88,000,000) exceeds the recreational lakes
48,000,000.6 The marginal efficiency of capital, however, is ambiguous since both
projects internal rates of return exceed the market rate of interest (assumed to be
equal to the social opportunity cost of capital). The Keynesian ranking criterion
would place the recreational lake ahead of the reservoir since the internal rate of
return of the former (9%) is greater than that of the latter (7%).7
Conclusion
I prefer the Fisherian marginal rate of return over cost and net discounted
^The assumption of costs being equal allows us to use the NDPV criterion on a
graph whose vertical axis is net discounted present value.
7 For the reader interested in further readings concerning the relative value of
the MEC and the MRRC see Robinson [1956], Alchian [1955], and Hirshliefer
[1958]. Robinson advocates the MRCCs assumption that the investments could be
renewed only at the market rate of interest after the expiration of the stated time
periods over the MECs assumption that the investments can be renewed on the same
terms. Hirshliefer [1958] examines the present value and internal rate of return
criteria. He concludes that MRCC works whenever the MEC does, and also
correctly discriminates among multiple tangencies when the capital market is perfect.
Even MRRC, however, fails in cases which involve the comparison of multiple
tangencies when there is an imperfect capital market. Multiple tangency cases result
from non-independent investment opportunities, i.e. where poorer projects are a
prerequisite to better ones. The failure of MEC in multiperiod analysis stems from
the implicit assumption that all intermediate cash flows are reinvested at X.
Additionally, MEC does not allow for varying interperiod preference rates.
Hirshliefers analysis of the investment decision in the two-period case can be applied
only to the simplest cases of multi-period investment.
30


present value approach to the Keynesian marginal efficiency of capital because under
similar assumptions, the marginal efficiency of capital approach is never significantly
simpler than the marginal rate of return over cost and net discounted present value
and often is more complicated. If we assume d = i, then the marginal rate of return
over cost and net discounted present value decision rules become simple criteria, just
as, under certain assumptions the marginal efficiency of capital is simple. If we
follow up on the distortions raised by Fisher and others which cause d ^ i, then the
marginal rate of return over cost and net discounted present value become
progressively more complex as the distortions mount.
Similarly, Keynes marginal efficiency of capital criterion increases in
complexity as we allow for real world distortions which make the determination of
the opportunity cost of capital more difficult. Additionally, the marginal efficiency
of capital increases in complexity as qualifiers are added to deal with situations which
cause the Keynesian approach to result in ambiguities. These ambiguities results
from three situations. First, situations involving negative cash flow sums, especially
when the initial year cash flow is positive, tend to result in ambiguity. Second,
ambiguity also occurs where there exists more than one internal rate of return for a
given alternative. Specifically, whenever the net annual cash flows during the n-year
analysis period are such that there are two or more sign changes, then the possibility
of multiple non-negative internal rates of return arises. Third, ambiguity tends to
32


occur in circumstances where a large end-year cost is accrued.
Additionally, Keynes marginal efficiency of capital criterion requires you to
determine the opportunity cost of capital which leaves you with virtually all the same
complications as the marginal rate of return over cost in addition to the marginal
efficiency of capitals complications. You may drop Keynes marginal efficiency of
capital and develop Keynesian internal rate of return criteria that do not require a
reference to the opportunity cost of capital. But these are simply ranking rules. For
example, A is preferred over B if the internal rate of return of A is greater than that
of B. But, the same can be done by dropping marginal rate of return over cost and
using net discounted present value ranking under various assumed and plausible
values of r. For example, A is preferred to B because A has a greater net discounted
present value than B until r is increased to 25%, which is assumed implausible. But,
if we require screening criteria like the marginal efficiency of capital and the
marginal rate of return over cost, or if we are concerned with optimality, then the
marginal efficiency of capital has basically all the same complications connected with
the marginal rate of return over cost (involving distortions to i through the
opportunity cost of capital) plus the additional complications necessary to correct
those ambiguities unique to the marginal efficiency of capital.
33


CHAPTER 2
DISTORTIONS TO THE MEASUREMENT
OF THE SOCIAL RATE OF TIME PREFERENCE
Introduction
In the Fisherian model, a perfectly competitive economy equates the social
rate of time preference and the opportunity cost of private capital. Fisher
complicated the model by introducing the distortionary effect of risk.
If distortions, such as risk, are not assumed away, then we need to either
adjust the market rate of interest or abandon it in favor of some alternate measure of
d. Four alternatives to the market rate of interest are used, both academically and
practically.8 These alternatives are the interest rate on government bonds, g, the rate
of return on private investment, v, the social opportunity cost of capital, 7, and the
social time preference rate, p.9 Generally, g, v, and 7 are used by those concerned
with distortions to the cost of capital while p is advocated by those most concerned
with distortions to the social rate of time preference.
8Eckstein [1957] and Eckstein [1958] provide a comprehensive discussion of these
alternatives, the interested reader is referred there.
9 The reader is referred to Chenery [1953] and Kahn [1951] who had earlier
applied alternatives to Fishers rate of interest into the cost-benefit analysis of
economic development projects.
34


Amartya Sen and Stephen Marglin head a school advocating the use of the
social rate of time preference as d. Their reasoning is based on the inability of the
isolated individual to express their desired rate of time preference for society in an
atomistic market. This school is initiated by Sens criticism of Ecksteins advocacy
of pure time preferences and consumer sovereignty. Eckstein [1957, p. 75] asserts
that, "a social welfare function based on consumers sovereignty must accept peoples
tastes including their intertemporal preferences." Sen argues that consumer
sovereignty is irrelevant to the question at hand given that consumers involved
include those in future generations.10
In criticizing pure time preference, Sen builds on Ramseys [1928, p. 543]
statement "that we do not discount later enjoyments in comparison with earlier ones,
a practice which is ethically indefensible and arises merely from the weakness of the
imagination." Sen [1961, p. 482] also draws from Harrods [1948, p, 40] assertion
that, "pure time preference [is] a polite expression for rapacity and the conquest of
reason by passion." Sen [1961, p. 482] summarizes his critique of pure time
preference and the choice between present and future consumption by saying that,
"while it is true that decision has to be taken now, there is no reason why todays
discount of tomorrow should be used, and not tomorrows discount of today."
1(yThe reader is referred to Sen [1961, p. 482].
35


Marglin builds on the arguments of Sen [1961] by examining the relationship
between individual p and d. Marglin [1963a, p. 96] asks, "is the concept of a social
rate of discount distinct from the individual rates of discount applied to unilateral
savings decisions a myth or a must?" Marglin discusses three solutions: the
"authoritarian," the "schizophrenic," and the "interdependent."11
The authoritarian solution answers Marglins "myth or must" question with
a "must," due to the "socially irrational" individual response to what Marglin [1963a,
p.96] terms the "brevity and uncertainty of life." Marglin attributes this line of
argument to Pigou. Pigou argues that government has a responsibility to represent
the interests not only of its current citizens but also of its future generations.
Governments ability to serve both is complicated by what Pigou argues is a socially
irrational attitude toward their own future on the part of members of the present
generation. Specifically, Pigou [1932, p. 25] states,
this preference for present pleasure does not the
idea is self-contradictory imply that a present
pleasure of given magnitude is any greater than a
future pleasure of the same magnitude. It implies
only that our telescopic faculty is defective, and that
we, therefore, see future pleasures, as it were, on
a diminished scale.
11 Marglin acknowledges the incorporation of individual time preference into v
resulting in 7. This relationship between v and 7 is discussed in Chapter 3. Marglin
concedes the point, but maintains that the incorporation of individual time preference
into measures of the opportunity cost of capital is irrelevant.
36


This characteristic distorts the time preference of individuals from social correctness
and causes the consumer rate of interest to be too high.12 To the extent this
irrationality inflates the value of present consumption at the expense of future
generations, government must reduce the marginal rate of discount below the market
determined rate. This new rate represents what some social welfare function deems
to be the correct value judgement.
Marglin contends that such a Pigovian social welfare function is fundamentally
in confrontation with the theory of democratic government. Marglin states that it is
"axiomatic" in democratic liberalism for government to reflect only the preferences
of its present citizens. For Marglin [1963a, p. 98], "a democratic view of the state
does not countenance governmental intervention on behalf of future generations."
Pigous solution is an example of a violation of the non-totalitarian assumption
of Arrows general possibility theorem. A judgement to lower the social rate of
discount below the rate reflecting the "defective" preferences of the unenlightened
majority would mean the imposition of Pigous or some other totalitarians
preferences over the majoritys.
The second answer to Marglins "myth or must" question is the
12 Both Baumol [1968] and Marglin [1963a] note that the Pigou-Ramsey answer
does not require that government necessarily invest more on the futures behalf, as
the answer only requires that total and not strictly public savings be raised.
37


"schizophrenic" or Rousseau-Colm solution. This line of reasoning, like Pigous
authoritarian solution, answers "must." This answer is not because the individual is
socially defective but rather because there simply do not exist unique individual time
preference maps. The preferences that determine an individuals economic (market)
actions are not the same preferences that determine an individuals political (social)
actions. As Marglin [1963a, p. 98] states "the economic man and the citizen are for
all intents and purposes two different individuals." Marglins reasoning follows from
Sen [1961, p. 487] who similarly states, "there is no reason to believe that a man
acting as a responsible participator in a political debate will express exactly the same
preferences as he does in day to day life." Hence the schizophrenia, and the
justification for the political rejection of market determined preference.
Rousseau first made the distinction between the general will and the will of
all. Rousseau [Book II, Chapter III, pp. 30-31] states, "there is often a great deal
of difference between the will of all and the general will; the latter considers only the
common interest, while the former takes private interest into account, and is no more
than the sum of particular wills."
Colm [1955] continues Rousseaus schism between economic man and the
citizen. Of Colm, Musgrave [1959, pp. 87-88] states,
Colm suggests... that there are political tasks in a
democracy that are only indirectly related to such
individual needs as are expressed in the market
38


place. ... He [Colm] holds that the individual voter
dealing with political issues has a frame of
reference quite distinct from that which underlies
his allocation of income as a consumer. In the
latter situation, the voter acts as a private individual
determined by self-interest and deals with his
personal wants; in the former, he acts as a political
being guided by his image of a good society. The
two, Colm holds are different things.
Colm [1955, p. 34] separates individual needs from the needs of the state. The
states needs are judged superior to all other ends and are limited only by the
individual needs necessary to maintain "human material with which the state must be
built." The states needs what is necessary to perform what Colm [1955, p. 34]
terms the "human ends of government...[which]...are individual and collective at the
same time."
However, the preference of the political over the economic is itself a value
judgement. The "schizophrenic" or Rousseau-Colm solution is merely a special case
of the "authoritarian" solution.
The third solution offered by Marglin is premised by the assumption that
individual utility is not only a function of individual consumption but also of the
consumption of others, both in ones own generation and in those to follow.
However, the market contains no mechanism for the expression of individual
preference for the consumption of others. That being the case, the individual seeks
a political solution. Through political mechanisms, the individual can wield the
39


coercive power of the state to force adherence of others to ones own preferences.
In the case of a majority voting rule, this may occur only by forming a majority
coalition with like-minded voters. In any other system, this may occur with the
exercise of sufficient political power by either an individual or a party of individuals
with similar preferences. Marglin differentiates this interdependence solution from
the schizophrenic by emphasizing that it is not concerned with changes in frame of
reference but rather in the mechanisms for expression available to the individual.
Marglin [1963a, p. 100] demonstrates the possibility of a Pareto superior
outcome, "by undertaking more investment collectively than each finds desirable to
undertake privately."13 Marglin sets up a simple two-period model in which all
current investment is universally available to a current generation of like-minded
individuals and bears all of its returns in a tomorrow inhabited exclusively by the
next generation. That is, everone alive today dies at days end, and the next adult
generation appears instantly on the scene tomorrow. We are asked to consider any
one individual, whose utility function can be expressed as,
ri(cPc/p-ci)
where q is the consumption of the ith individual, cf is the consumption of the future
13The reader interested in this line of reasoning is further referred to Sen [1961]
and Baumol [1952].
40


generation, and cp is the consumption of the individuals own generation. Taking
partial derivatives we find,
8c,
that is, as Marglin [1963a, p. 101] states, "the marginal utility of ones own
consumpsion is unity." Additionally,
and
duf
3(cp-cf)
That is, a is the marginal utility the individual receives from consumption by the next
generation, and /3 is the marginal utility the individual derives from consumption by
members of the individuals own generation other than themselves. Marglin adds to
this one additional parameter, the marginal rate of transformation, k. With these
parameters, Marglin [1963a, p. 101] states the rule for unilateral individual
41


investment as, ak > 1 (2.1)
i.e. invest if the marginal utility derived from future consumption times the marginal
rate of transformation is greater than or equal to unity.
Next, Marglin looks at how investment by others of the individuals
generation affects that individual and finds that: if the marginal utility derived from
future consumption times the marginal rate of transformation is equal to or exceeds
the marginal utility derived from the consumption of others in the individuals
generation other than himself, then the individual is made better off. In other words,
if
ak > /3 £9
then the your investment makes me better off.
Marglin assumes values of a = 1 (that is the individual is indifferent as to
whether or not to sacrifice ten cents of their own consumption for one dollar of future
generation consumption), /3 = .15 (individual indifference between not forgoing and
foregoing fifteen cents of their own consumption which would allow one dollar of
consumption to others in their own generation), and k = 2 (a marginal rate of
transformation from current to future consumption of two). Marglin has made the
individuals in his model all rather generic in terms of preferences. All individuals
are guided by essentially the same parameters applied under the unilateral investment
decision rule (2.1). Unless we assume super-altruism, it is unlikely that anyone
42


would undertake unilateral investment even though all would like to see other invest,
as expressed by equation (2.2).
For the sake of observing the effects of different parameters, let us assume
here, an even greater altruism and marginal rate of transformation, such as, a = .2,
/? = .5, and k = 3. Marglin has presented us with a decision rule for unilateral
investment and a criterion under which the individual benefits if others invest. Next,
we are presented with the decision rule under which each individual is willing to
invest provided everyone, n, also does also. Marglin states this rule as,
akn > 1 + 0(n 1). (2.3)
Marglin [1963a, p. 103] explains this rule by stating,
each is made better off so long as the marginal
gross gain from investment of n dollars (Marglin
has assumed that each individual invests one dollar],
akn, exceeds the loss of utility from his personal
sacrifice [1] plus the psychic loss he feels from
others sacrifice, 1 + /3(n-l).
For the parameters assumed here, akn > 1 + (3(n 1) when n > 5. That is, for
any community with a population of five or more, and under the values set for
parameters a, j3, and k, the collective investment rule will hold.
Marglin suggests we examine this problem within the structures of a game
theory matrix, although he does not provide such a matrix in his article. A matrix,
based on the marginal utility functions provided by Marglin, is given in Table 3.
43


Marglin [1963a, p. 101] defines the marginal utility function as,
U.=c.+acf+$(cp-ty,
where A denotes an infinitesimally small change. Additionally, Marglin [1963a, p.
103] states,
-1+akn- p (n-1).
This equation represents what Marglin [1963a p. 103] calls the "change in utility
from the point of view of each if everyone invests one dollar." This equation is used
in the collective solution of Table 2, where both the individual and all, n 1, others
invest. The equations in the other solutions derive from Marglins marginal utility
function. The assumption of n = 100 is sufficiently high to satisfy akn > 1 + /3(n
- 1), which is the condition Marglin [1963a, p. 103] requires for each of us to be
willing to invest provided everyone else does also. Note, the exception in game 3
where no value for n is sufficiently great for it to hold.
In games 1 and 2, we find the matrix results implicit in Marglin. That is,
given Marglins parameter values in game 1 and the second games greater altruism
and marginal rate of transformation, we find that everyone is made better off in the
collective than in the individualistic solution.
However, as game 3 demonstrates, this games outcome is unstable. Minor
changes in parameter values can result in a matrix where the maximized minimum
44


TABLE 2
Others (n-1) Invest Other do not Invest
Individual The Collective Solution The Unilateral Solution
Invests akn-[B(n-1) + 1] -[ak(n-1) + 1l+ak+B(n-1)
Individual does The Parasitic Solution The Individualistic Solution
not Invest [ak(n-1) + 1l-a k+B(n-1) -akn + rB(n-1) + n
SOLUTION SET
The Collective Solution The Unilateral Solution
GAME 1 4.15 (4.75)
GAME 2 9.50 (9.30)
GAME 3 (6.80) 6.08
GAME 4 (1.00) 0.40
GAME 5 (0.01) (0.57)
The Parasitic Solution The Individualistic Solution
GAME 1 4.75 (4.15)
GAME 2 9.30 (9.50)
GAME 3 (6.08) 6.80
GAME 4 (0.40) 1.00
GAME 5 0.57 0.01
a B k n
GAME 1 0.100 0.150 2.000 100
GAME 2 0.200 0.500 3.000 100
GAME 3 0.070 0.200 2.000 100
GAME 4 0.100 0.202 2.000 100
GAME 5 0.110 0.200 1.890 100
45


occurs in outcomes other than Marglins collective solution, game 1. In games 4 and
5, we have a situation in which the cooperative solution is never optimal. In game
3, we found that each individual will (in order to maximize their minimum) choose
the -0.08 for not investing over -1.9 for investing. Therefore, no one invests and
every individual as well off as the game allows, the individualistic solutions 1.9.
In game 4, with parameter values of a = 0.1, /3 = 0.202, and k = 2, if the
individual invests that person is made worse off regardless of what everyone else
does, and likewise that individual is made better off by not investing regardless again
of others. In game 5, with parameter values of a = 0.11, j3 = 0.2, and k = 1.89,
the individual is indifferent between everyone investing and no one investing but the
game will result in no one investing as all attempt to avoid the unilateral solutions
-1.58.
At the outset of his paper, Marglin states that he is only trying to demonstrate
that it is possible for a game to exist which results in the individualistic (market)
solution even though a Pareto superior outcome is possible by the government forcing
us to the cooperative (collective) solution. Given the parameter values he uses (i.e.
game 1), he does prove it possible, although equally plausible parameter values
demonstrate it is not the necessary outcome.
Expanding the discussion of games and marginal utility functions, Marglin
presents us with the two implicit discount rates. The marginal private rate of discount
46


for the ith individual is,
and the marginal social rate of discount is,
j-i+fr-DP
' an
The values for each game of the social and private discount rates are given in Table
TABLE 3
Game
1 2 3 4 5
r i 9.000 4.000 13.286 9.000 8.091
d i 1.585 2.525 2.971 2.100 1.891
It is surprising to me that game 2, which results in the greatest value for the
cooperative solution, has a higher marginal social rate of discount than game 3,
which resulted in the lowest value for the cooperative solution.
47


Conclusion
Sen [1982] presents three lines of reasoning in favor of d = p < y. First,
"super-responsibility" contends that government is responsible to future
generations "over and above" the present generations preferred sacrifice.
However, Marglin [1963a] states that under democratic liberalism it is
"axiomatic" that government only reflect the preferences of the current generation.
I find little merit in Sens super-responsibility and side with Marglins
axiom. Similarly, I reject the authoritarian solution as a totalitarian impositions of
the will of an elite over the preference of the majority.
Second, Sens "dual-role" argument maintains that individuals act in two
frames of reference, that of the citizen and that of economic man. This is the
"schizophrenic answer" of Marglin [1963a]. Whatever its called, the imposed
preferencing of the citizen over the economic man is itself a merely a form of the
authoritarian solution.
Third, the "isolation" argument initiated by Sen [1961] and extended by
Marglin [1963a] and Sen [1967] emphasizes the non-optimality of market saving
through an "isolation paradox." This paradox is an n-case extension of the
prisoners dilemma under which Pareto-inferior game solutions result. Sens
isolation paradox is demonstrated in Marglins individualistic and collective game
solutions. This argument concludes that individuals have a lower rate of time
48


preference than they express individually because their isolation precludes actions
in concert with others in society.
There are several strong criticism of Marglins and Sens arguments for a
social rate of discount equal to a social rate of time preference which is less than
the market rate of interest. Marglin and Sen fail to address the possibility that an
individual may make an investment motivated by concern for the future, and at the
same time make a gift of charity for the presents poor. Baumol [1968, p. 800]
states, "we must then ask ourselves whether there are so few diseased, illiterate,
underprivileged today, so few persons who excite our sympathy that we must look
to the prospectively wealthy future for a source of worthy recipients." Baumol
[1968, p. 800fn] continues this line of criticism stating, "it is my judgement that
the probable wealth of future generations is given inadequate weight... so that the
Marglin-Sen ... externalities may well prove benign in their effects."14
Marglin presumes that the optimal rate of investment exceeds the rate
which would be chosen when each person decides individually how much they
would like to save for the future. This is due to the assertion that each
individuals charity has a negligible impact on societys consumption in the present
or in the future. Consequently, we may assume, that the individual maximizes
14 The reader, further interested in this line of criticism, is referred to Tullock
[1964] and Baumol [1968].
49


only the hedonistic portion of their utility function. However, if the individual
believes that society is over or under investing, that individual has recourse to
public action designed to induce more or less investment.
Marglins argument indicates that the optimal rate of investment may be
greater or less than the rate resulting from individual decisions. As evidence of
future-oriented altruism, Marglin cites the general concern for economic growth.
However, even if an individual wants economic growth, that individual may not
necessarily be prepared to buy it by reducing present consumption. Lind [1964,
p. 345] states,
if each individual, given the market equilibrium,
feels that the marginal rate at which he is willing
to trade present for future consumption is correct
for every other individual, then his private and
social rates of discount are equal and the market
rate of interest is of normative significance for
public investment decisions. In a society whose
members have similar preferences and incomes, it
is reasonable to assume that this holds.
Neither the theoretical argument nor the empirical evidence establishes that
the optimal rate of investment necessarily exceeds the rate chosen by a collection
of individual decisions.15 However, to the extent that the individuals impacted
by a project have dissimilar preference and incomes, there is increasingly less
15The reader interested in this critique is referred to Usher [1964].
50


reason to suspect that the market rate of interest will reflect the social rate of time
preference. An evaluation of a particular project needs to consider the relative
similarity or dissimilarity of those affected. If it is suspected that the social rate
of time preference does diverge from that reflected in the consumers rate of
interest, then the best option available to us is to attempt to determine the social
rate through the political process.
51


CHAPTER 3
DISTORTIONS TO MEASURES OF THE MARGINAL RATE OF RETURN
ON PRIVATE INVESTMENT AND THE SOCIAL OPPORTUNITY COST
OF CAPITAL
Introduction
Marglin and Sen maintain that for several reasons d = p < i. Similarly, a
second school argues that various distortions cause the social opportunity cost of
capital to deviate from the market rate of interest. Two distortions in particular are
focused upon in the literature, those resulting from risk and those resulting from the
corporate income tax. Contrary to the Marglin-Sen school, this school generally
advocates d > i. We find three alternatives to i advocated: the interest rate on
government bonds, g; the rate of return on private investment, v; and the social
opportunity cost of capital, 7.
The Interest Rate on Government Bonds
The rate of interest on government bonds, g, is the most pragmatic alternative
to Fishers equilibrium market rate of interest simply because it is easily measured
and widely understood. In practice, the use of indices based on the interest rate on
government bonds is common. There are numerous criticisms of g as d. First, the
credit risk to the lender of funds to the government is small. Eckstein [1958, p. 96]
52


points out that "the federal bond rate does not measure the full social cost of capital
since it makes no allowance for the risks of the project."
Second, the use of bond rate neglects the nonvoluntary taxation often required
to fund public projects. Implicit in the use of g is the assumption that all projects
being evaluated are financed by the sale of bonds and not through taxation. Allowing
for taxation causes g to become much less meaningful.
Third, government and society are not synonymous. The rate of return on
government bonds is a measure of the governments cost of capital; it is not
necessarily a meaning of social cost.16
Fourth, there appears to be no agreement on which bonds or bundle of bonds
to use. The rate on any bond, even bonds with maturities matched to the cashflow
of the evaluated project, is a function of numerous variables which may be irrelevant
to a measure of the social opportunity cost of capital. Additionally, bonds are priced
so as to generate the expected total return necessary to compensate the investor for
expected risk. Further, the expected total return and g are much more a function of
a bonds duration than its maturity. These points are illustrated in Figure 3.1. In
this graph, the government bond market exhibits a highly inverted yield curve
structure.
16The reader interested in this line of reasoning is referred to Feldstein [1964a,
p. 121].
53


Figure 3.1 is indicative of a market that has built in an expectation of
declininginterest rates. The elasticity of total return on a bond with respect to
changes in interest rates increases as the duration of the bond increases. Therefore,
the lower current interest rate on longer duration government bonds may actually
mask a higher expected total return, given the expected change in interest rates.
Therefore, g is a speculative component of the expected total return.17
FIGURE 3.1
The second suggested measure of d is the rate of return on private sector
17 Baumol [1968, p. 797fn] touches on expected return indirectly, suggesting that
g = rp where r is some weight and risk premium associated with an expected rise
in prices (inflation).
54


investment, v.18 Eckstein [1958, p. 97] states (but does not advocate) the case for
v as,
capital employed in public projects could be used in
fields of private investment instead, where it would
earn a high rate of return, and therefore ... an
optimal allocation of investment in the economy
would require that the rate on public investment
should be as high as the rate in the private sector.
The main proponent of d = v is Hirshleifer et al. [1960, p. 139] who states, "if
funds were not invested in the public project, they would have been used in private
investment. The proper rate of discount is therefore the rate of return on private
investment."
There are a several configurations of v. However, the marginal rate on
similar investments is preferable for two reasons. First, the marginal rate takes
account of diminishing returns. Second, the provision of projects similar to the
proposed is preferable because these projects are affected more similarly by
distortions than dissimilar projects.
There are three major problems with v = d. First, serious distortions
18 There are advocates of g as a measure of the rate of return in the private
sector, v. Which in turn assumes v to be important. If one argues that g
approximates or is assumed equal to v which in turn approximates d why not save a
step or an assumption and advocate v? Feldstein [1964a, p. 122] argues against the
assumption of g = v, stating, "a failure to recognize (that g < v) is the basic error
of those who equate (g) with the ex post (i.e., after allowance for losses) rate of
return on private investment."
55


resulting from risk and taxation need to be rectified by any measure of v. Second,
even a risk- and tax-adjusted marginal rate of return on similar private sector
investment fails to account for the social costs and benefits resulting from
externalities (as does the rate of return on private investment). Feldstein [1964a, p.
125] identifies one of these externalities as "increases [to] the productivity and
therefore the earnings of other factors of production." It is likely that the social
marginal rate of return on private investment, v, is greater than Hirshleifers rate.
Second, both v and v assume that funding for public projects is diverted
entirely from private investment. This is a weak assumption which ignores the
likelihood that public investment funded by taxation or debt diverts funds from
private consumption. Because v fails to consider the social rate of return and the
likelihood of consumption being diverted to fund public projects, Eckstein [1958, p.
97] declares it to "hold little normative significance for public projects."
The Social Opportunity Cost of Capital
The social opportunity cost of capital, 7, is the social cost of foregone private
consumption incorporated into a tax and risk adjusted v. Baumol [1968] deals with
the adjustment of v for risk and taxation. Baumol is not advocating that v = d.
He advocates the social opportunity cost of capital, 7. Rather, he is addressing
distortions to that component of 7 which is v. Baumol [1968, p. 790] advocates that
56


d be formulated so that, under a constant level of employment, it will "lead to a
positive number for the evaluated net benefit of a public project if and only if its
gross benefits exceed its opportunity costs in the private sector."
Baumol sets forth a simplistic model for his analysis, assuming:
1) a fixed level of employment of all resources in the
economy
2) no risk or uncertainty
3) a private sector comprised exclusively of corporations
4) corporations financed entirely through equity
5) a uniform corporate income tax rate of 50%
6) a unique interest rate, g, at which the government
borrows.
When considering some public project requiring the use of a set of input resources,
R, for some given period of time, the government must obtain R by forcing its
relinquishment by the corporate sector. The opportunity cost of a public project may
be determined by calculating the returns which would have been obtained had R
remained in the private sector.
This calculation of opportunity cost begins by considering the situation facing
corporations. In order to attract the marginal investor, a corporation must pay its
shareholders a rate of return at least equal to the rate of interest paid by the
government on its bonds, g. However, because corporate income is subject to
taxation prior to the distribution of dividends to its stockholders, the corporation must
earn a rate of return of g/(l-t). Since the corporate income tax rate, t, is assumed
57


to be 50%, then g/(l-t) = 2g = v. Thus, corporations must earn two times the rate
on government bonds in order to give their shareholders, a return equal to g, which
is net of the corporations income tax liability. Baumol has exposed a "wedge"
driven between g (and also the return on corporate stock) and v, the rate corporate
investments must earn in order to pay their investors g. The cause of Baumols
wedge is the corporate income tax.
Baumol also considers two consequences of risk. The first consequence is a
substitution of debt securities for equity securities in order to attract investors
interested in limiting risk. The second consequence is the risk premium paid by the
private sector to create investor indifference between their own securities and the
relatively safer government bonds.
Baumol relaxes his assumptions to allow corporations to raise capital through
equity and debt. Interest payments on debt are not subject to the corporate income
tax (it is in actuality an expense write-off against corporate income), nor are (with
the same effects) earnings of noncorporate firms. It follows that the previous
estimate of v is overstated and that this estimate needs to be reduced. Baumol [1968,
p. 793] suggests that one should decrease,
the figure of [v = g/(l-t)] for the corporate sector
perhaps proportionately to its use of debt financing
and then the overall discount figure should ... be
reduced in proportion to the resources that would
come from noncorporate enterprises.
58


Baumols definition of v becomes
(l-c)g+c(il-e)g+e(gl(l-t)))=g+ceg,
where t = 0.5. Thus, if we set the corporate sector portion of GNP, c, at .6, and
set the portion of corporate financing accounted for by equity, e, at .5, then v = g
+ ceg = 1.3g. That is, v is 1.3 times the government bond rate of interest.
Baumol is more concerned with the second consequence of risk, the risk
premium. Common sense tells us that, to the extent a project entails risk, society
should further discount a potential project over and above that pertaining in a riskless
rate. Such a project must then return an additional risk premium relative to the less
risky project to create societal (or individual) indifference.
However, from societys point of view the expected rate of return on its whole
portfolio of numerous public and private investments becomes virtually certain.
Baumol [1968, p. 795] states that
with increasing numbers of [private] projects the
marginal value of this sort of risk, too, will be
negligible. Thus, from the social point of view the
law of large numbers argument cuts both ways-it
says that risk in either the public or private projects
is irrelevant for the returns society can expect.
59


This leads some to argue for the exclusion a risk premium from d.19 For, as the
public sector engages itself in larger and larger numbers of varied projects, the
overall outcome of these projects becomes known at the aggregated level of all its
projects expected values. In other words, society should be indifferent between two
projects offering two possible payoffs with equal probability because both projects
would have equal expected values.
For example, two projects might offer payoffs with equal probabilities of 0
or 200 and 99 or 101. However, both projects offer an expected payoff value of
100. One might argue that society should be indifferent towards each even though
the former encompasses significantly greater risk relative to the latter. Therefore, no
risk premium should be incorporated into d.
However, Baumols paradox [1968, pp. 795-766] argues, not for exclusion
of the private sectors risk premium, but for its inclusion:
does this mean that the risk discount component in
private cost of capital figures should be ignored in
the social rate of discount calculation, as is often
suggested in the literature? On the contrary,
paradoxically, the very absence of real risk means
that the private risk discount should also enter the
social discount rate. Here private risk plays
precisely the same role as the corporation tax. It
induces firms to invest in such a way that the
19The reader further interested in this reasoning is referred to Samualson [1964]
and Arrow [1966].
60


marginal investment yield is higher than it would
otherwise be. And the transfer of resources from
the private sector therefore imposes a
correspondingly high opportunity cost.20
an
ag
ina
Baumol recognizes that private sector investors engage in risk-pooling. That is, if I, as
investor, diversify my portfolio with larger and larger numbers of holdings, then the expected
regate value of my holdings becomes more certain, and the risk premium which I require to
i ice me towards indifference between the marginal private and public sectors investments
app roaches zero. Additionally, through mutual funds, managed pension funds, index investing,
and pooled mortgage pass through securities, the individual can quite easily develop a diversified
por folio. To the extent that the private sector investor does engage in diversification of
holdings, Baumols risk premium is significantly reduced.
Baumol notes two additional considerations which may require adjustment to the risk
premium. The first is that the incursion of risk by the private sector constitutes a disutility to
20Pauly [1970, p. 195] concurs with Baumol, stating, "private aversion to risk
causes (v) to diverge from the rate at which individuals are willing to loan money to
the government (g). ... necessarily understates the true social cost of the investment,
and so should not be used as a measure of the social rate of discount. ... Hence,
divergence between (g) and (v) does not necessarily indicate the absence of an
optimal rate. ... if taxation-induced distortions are ignored, the "correct" rate for
evaluating government investment opportunities may be equal to, or greater than, the
private rate (v)."
61


the investor, a psychic cost which has no counterpart in public investment.21 This point fails
to < onsider that those responsible for public projects also incur a disutility contemplating the
pro ;pects of their projects failing. Otherwise, this hardly constitutes a reason to give the public
sector managers more resources not to care about. Also, if this is so, why should not there also
;. psychic entrepreneurial premium an additional utility incurred when those in the private
be
sect or contemplate their risky projects succeeding, which has no counterpart in the public sector?
At lj)est, this assertion is a wash.
The second consideration is corporate insolvency. Baumol [1968, p. 795] states that "a
private firm faces some danger of insolvency, in which case a project which has been undertaken
maj[ never be completed." Baumol critiques this assertion by suggesting that this may be offset
by tjhe danger of political insolvency, that is, a change in government which may leave public
projects uncompleted. Additionally, today there is the reality of Chapter 9 bankruptcy and
pub ic-sector insolvency.
Assume a project conducted by a corporation which, after an initial investment and for
exogenous reasons, becomes insolvent. The inability of the corporation to continue funding this
project would suggest imperfect capital markets. If we are allowing for imperfect markets
shd\ Id we not also allow for imperfect central planning? If the corporate insolvency is due to
som; other event, for example losses incurred in another corporate project, would not some
21 This consideration is credited by Baumol [1968, p. 795fn] to a letter from
William Whipple of Rutgers University.
62


oth ;r investors be interested in continuing the project? It is at least as likely that a public project
would remain uncompleted as a result of political uncertainty or changes as it is that a private
sector project would remain incomplete for economic reasons.
Baumol [1968, p. 796] sums up his discussion of v by concluding that "it is irrelevant
to alrgue that (v > g) is produced by artificial distortions taxes, risks which for society do not
exis t, etc. The fact that the source of (v) is artificial makes the resulting yield figure no less
sub itantive." However, v = y = d results from the assumption of a private sector made up
entirely of corporations. Introduction of consumers into the private sector along with taxes
whi|sh reduce consumption lead us to consider the second component of the social opportunity
coSfl of capital, foregone consumption.
Krutilla & Eckstein [1958] were the first to advocate a weighted average of the rate of
interest at which consumers borrow, x, and v. If we define "a" and "b" as the relative weight
of capital diverted from consumption and private investment respectively, then
d = 7 = ax + bv where a + b = 1.
Thbl methodology for determining the weights becomes increasingly complex as models are
generalized to allow for various forms of taxation and manifestations of risk. The important
concept is the consideration of the incidence unique to each project.22
22 The reader further interested in this method is referred to Tullock [1964], Lind
[1964], Feldstein [1964b], Usher [1964], and Ramsey [1969] all argue for d as some
weighted average of y and p.
63


Criticism of the social opportunity cost of capital school centers on the pitfalls of
averaging x and v or 7, and on the assumption of a two-period case. Under Fisherian
equilibrium, social optimality occurs when 7 = p = i = d. Only if these two terms are
eqp ivalent can the single term d be applied. When the two rates are not equivalent, methods of
ave raging are inadequate in preserving all relevant information. Averaging is inadequate because
th^ie is an infinite set of coordinate (7,p) solutions.23
Diamond [1968] uses a two-period model to, like others, reach an "intuitively appealing
reiiflt," which is 7 = d. Diamond [1968, p. 683], however, is well aware of the hazards of this
approach and characterizes the two-period cases results as "extremely misleading in the more
jral case." By using the two-period approach, one has eliminated a serious complexity of
problem of re-investment of future benefits. A method is required to capture multi-period
3 acts of foregone or created net benefits. This method is the shadow price of capital
approach.
gen
the
1m
23 The reader interested in this topic is referred to Marglin [1963b] and Feldstein
[1972], who show that if the artificial two-period assumption is relaxed, the case for
using a weighted average of the two rates breaks, down. The reader is further
referred to a number of "divergence" articles. Pigou [1932] suggests a public
investment decision criterion based on the ability of a project to reduce the difference
between 7 and p. Feldstein [1964b] and Marglin [1963b] dismisses this criterion as
unworkable. Also one may consider a decision criterion where a project is accepted
if its benefit/cost ratio, discounted to present values by the p, is greater than yip.
64


CHAPTER 4
THE SHADOW PRICE OF CAPITAL APPROACH
Capital market imperfections, such as distortionary taxation and risk, cause y and p to
di^irge. If a project discounted at both rates has benefit-cost ratios greater than one, then we
wo\ld accept the project and the divergence is not problematic. Likewise, if the both ratios are
beldw one, we reject the project. The difficulty occurs when the ratio is less than one when
discounted at one rate (typically 7, since it is often argued that 7 > p) and greater than one
when discounted at the other rate. On the one hand, there is an opportunity for a welfare gain
when a project pays more than p; on the other hand, it may be possible to do better by having
thes|e funds in the private sector at 7.
Lind [1982 p. 39] states that in the shadow price of capital method, "[d] is set equal to
[p] and the effects on private capital formation are accounted for using the concept of the
shadow price of capital." Put another way, Lind [1982, p. 41] argues that the correct method
of public investment analysis is to,
trace the impacts on consumption over time and
then to discount at the social rate of time
preference. ... [To trace the impact on
consumption] compute the shadow price of capital
and then multiply the costs of public investment that
represent a displacement of private capital by the
shadow price to obtain the true opportunity cost in
terms of consumption.
Lind [1982, p. 39] defines the shadow price of capital as "the present value of the


stream of consumption benefits associated with $1 of private investment discounted
at the social rate of time preference." Similarly, Bradford [1975, p. 889] explains
the shadow price of capital in this manner:
imagine that a unit of time t capital drops like
manna from heaven, affecting the whole stream of
consumption starting at time t+1. In this new
consumption stream, by how much at most could
one reduce the amount of consumption in period t,
and still have a consumption stream as valuable
[according to government preferences] as the
original, pre-manna, consumption stream? The
answer is 8, and in this sense, 8 is the social value
of a unit of private capital at time t.
As discussed, some of the literature suggests that p = x. Others argue that x fails
to consider intergenerational equity problems as well as distributional issues within
a generation. Stiglitz [1982] demonstrates that the shadow price of capital may be
calculated directly into the social rate of discount, in which case a distinct discount
rate would be generated for each project. The notion, once universally held, of a
single rate of discount is no longer believed valid.
The development of the shadow price of capital method began with Dasgupta
et al. [1971]. Dasgupta et al. [1971] assume public investment benefits of B per
year, starting in year 1 and continuing forever. The present value of the stream of
benefits, discounted at the socially optimum discount rate p, is BIp. All capital costs
accrue in year 0 and equal K. If all funds are taken from consumption, the
66


appropriate criterion would be to invest if BIp K >0. Were all funds are taken
from investment, the new criteria would be to invest if B/v K > 0. If public
investment displaces both private consumption and investment, then the criterion is
to invest if BIp (yip)K > 0.
The quantity v/p is the shadow price of funds drawn from private investment,
P. This may be alternatively thought of as the adjustment in capital costs needed to
express K in terms of consumption units when society is saving at the optimal rate
and when v does not equal p. Therefore, the investment decision rule may be
expressed as,
B/p [cP + (1 c)]K > 0,
where c is the proportion of output devoted to private investment or the proportion
of public investment funds that displaces private investment, and P is the shadow
price of private investment.24 For example, Table 5 uses the Dasgupta shadow price
of capital decision rule to evaluate the four similar projects.
Table 4 illustrates the sensitivity of the decision rule to small changes in the
values assumed. This rule is a screening criterion using the shadow price of capital.
A similar presentation of the shadow price of capital method is given by Bradford
[1975]. Bradford replaces Dasgupta et als c with s, the marginal propensity to save
24The reader further interested in the development of the shadow price of capital
method prior to Lind [1982] is referred to Sugden & Williams [1978].
67


TABLE 4
Variable Project A Project B Project C Project D
B $1,000.00 $1,000.00 $900.00 $1,025.10 Social benefits per year forever
K $20,000.00 $20,000.00 $20,000.00 $20,000.00 Capital costs, all accruing in year 0
V 8.5% 9.0% 8.5% 9.0% marginal rate of return on private capital
4.6% 4.9% 4.6% 4.9% social rate of time preference (equal to x)
c 5.0% 5.5% 5.0% 5.5% proportion of output devoted to private investment
Dasgupta decision rule, invest if B/ [cP + (1 c)]K > 0
For Project A,
B / - [cP + (1 - -c)]K = $891.30
For Project B,
B / - [cP + (1 - - c)]K = ($512.24)
For Project C,
B/ - [cP + (1 - -c)]K = ($1,282.61)
For Project D,
B/ - [cP + (1 - - c)]K = $0.00


out of disposable income. Bradford develops a measure of the shadow price of
capital, 5, and incorporates it slightly differently than Dasgupta et al. Bradford
discounts the benefit and cost cashflows by p and then adjusts the discounted cost
value by multiplying it by 5. What results is a decision criterion which is a
discounted ratio of benefits to costs incorporting 5. Bradford [1975, p. 894] states
that
where

(l~s)H
1-S|1
(1 +
(1+P)
v is the marginal one-period private sector rate of return on capital, and p is the
social rate of time preference corresponding to a unique level of consumption and
under the assumption of constant coefficients p and s.
The effect of different parameter values on the shadow price of capital is
illustrated by Bradford. Table 6 illustrates these effects in a manner similar to
Bradfords, but with different parameter values. Here the shadow price ranges from
a minimum of 0.98 at i = .09, s = .05, and r = .07 to a maximum of 1.18 at i =
.03, s = .15, and r = .18.
69


TABLE 6
The shadow price of private capital under a range of parameter values
s = .05 s = .15
r .07 .12 .18 .07 . .12 .18
i .03 1.04 1.09 1.15 1.05 1.10 1.18
.06 1.01 1.06 1.12 1.01 1.07 1.14
.09 0.98 1.03 1.09 0.98 1.03 1.10
Lind [1982] reworks Bradford [1975] and establishes the shadow price of capital
decision criterion. Lind incorporates Bradfords definition of 5 into a screening type
decision rule. Linds rule is to invest if,
Uq l+g(6-l)^0
1+p l+a(6-l)
where 5 is as defined by Bradford.
The shadow price of capital decision criterion works best when p=x.
However, this condition may not always hold. Stiglitz [1982] identifies the
"constraints" created in combinations specific to the nature of each project. These
constraints cause the divergence of p and x as well as other rates. Stiglitz [1982, p.
187] identifies the following constraints:
A. No head tax;
B. 100% pure profit tax not allowed;
C. Restrictions on governments control on real money supply (debt);
D. Restrictions on the governments ability to differentiate tax rates (e.g.,
70


different types of labor must be taxed at the same rate);
E. Restrictions on the governments ability to choose the tax rate.
These five constraints combine to create the divergences among the rates
which are equal under "first-best" conditions. Table 7 illustrates the substance of
Stiglitz contention.
TABLE 7
First-Best (no constraint) d = x = v = p
Second-Best (one constraint) Under Constraint A d = v = p 5* X
Third-Best (two constraints) Under Constraints A,B Under Constraints A,C Under Constraints A,D or E d = V = p 9* X d = p, d need not lie between x and v d ^ p, v ^ p
Fourth-Best (three constraints) Under Constraints A,B,C Under Constraints A,B,D or El Under Constraints A,C,D or EJ either p Fifth-Best (four constraints) Constraints A,B,C,D or E d need not lie between x and v, or v and p; different d for different projects.
The problem is to determine the constraints which are relevant to the
particular project being evaluated. The result will be a different rate for each project.
Lind [1982 p. 443] states, "I believe that there is no single rate," and Lind [1982 p.
71


10] concludes "the advances in theory have led to the belief that perhaps different
rates are appropriate for different projects and policies."
Conclusion
The most compelling answer to the thesis question (what does economics
suggest as a rate of discount for those evaluating public projects?) is found in the
shadow price of capital decision criterion as laid out by Lind [1982] building on
Bradford [1975]. Stiglitz [1982] and Marglin-Sen all have demonstrated the difficulty
in determining p. Unfortunately, p is an integral component of the shadow price
criterion. However, the divergence between x and p may not be significant enough
to totally destroy the utility of the criterion using x as a measure of p.
The economics literature offers the analyst of public investment a good method of
getting a screening of proposed projects. From there, the determination of project
selection must remain a political process, but in all likelihood the use of the Criterion
will protect economic efficiency from being severely damaged.
Economists need to continue the work on measuring the social rate of time
preference under more difficult constraints. Lind [1982, p. 449] incorporates
conditions typical of government investment projects into the shadow price of capital
criterion and finds d = 4.6%. To the best of my knowledge this is the best answer
the literature has to offer the analyst in search of a rate at which to discount.
72


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and the Choice of Discount Rate," American Economic Review, 65 no.5, 887-
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