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# Cost-Benefit Analysis

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• pg 1
CBA 100401

Discounting, Inflation, and the Social
Discount Rate

• Most material in Chapter 6 in text is straightforward and just
make sure you know it. I will just review some important
points of the general material
• In Ch 15 in the Pindyck-Rubinfeld text, discounting is
discussed in the context of the markets for factors of
production: “capital” is one of the two main factors of
production
• In simple statice macroeconomic models one thinks of the
price of capital as a “rental rate”. But one can also interpret
the price of capital as the rate of interest that is determined in
the market for savings and investment
– in some simple static models, the amount of capital is taken as given
from the outside. However, once we recognize that capital is a
produced factor of production, we must consider how it is supplie

• The supply of capital is determined by the willingness of
income earners to help finance the production of capital
through savings. Hence in the long run its price is determined
in the market for savings and investment
• Chapter 10 on how to determine the Social Discount Rate to
be used in CBA of long-lived government projects is very
important, and we will talk about that more later. One
important determinant of how the social discount rate is
related to observable market interest rates is inflation, and we
will discuss inflation measures at some length today; it is
discussed somewhat in the text, but more in the Journal of
Economic Perspectives article by Moulton, which is available
through the CUFE library.
The basic discounting formulae

• A general formula for computing the NPV of a project with
costs and benefits over many time periods (which should
measure the sum of the CVs), is

NPV  t 0 wt ( Bt  Ct )
T

where the       wtare discount factors. By convention,

the discount factor at time zero (“the present”) is 1. In general, if
the rate of interest r is allowed to be different in each time
period, we also have

wt 1  wt  (1  rt ) 1
Basics, cont’d

• In the special case where the value of r is constant in each
year, we will have
( Bt  Ct )
wt  (1  r ) , NPV  t 0
t                    T

(1  r ) t
• Make sure you know the exact meaning of the summation
notation
• Note that in some discussions authors start from t=1. This of
course is just a notational convention, but make sure that you
know how to modify the formula and interpretation when
that is so
Basics, cont’d

• One point that is made in the text is that in contexts where
some of the B and C are flows that occur evenly over time, the
ones that occur in the first year (year 0) should appropriately
be discounted to the midpoint of the year. Thus if all B and C
are flows, you could change the formula to

( Bt  Ct )
NPV  t 0
T

(1  r ) t  0.5
• If you want to take account of the fact that interest can be
compounded more frequently than annually (in some
contracts it is done monthly or even daily, you can compute
NPV by summing over shorter intervals (say, months)
Discounting, cont’d
• Specifically, suppose the project lasts for 10 years (T=10), but
you want to compute the NPV using monthly compounding of
interest; suppose the discount rate is 6% per year. You can
then do so by defining each unit of t as one month. You then
estimate monthly flows of benefits and costs, and compute
the NPV as the sum of 120 terms (rather than 10). Your
discount factors now become

wt  (1  (. 06 / 12 ))  t
where t is measured in months.
• In the limit, with continuous compounding, the formula
becomes
wt  e  rt
and the NPV is computed as an integral of B and C flows
“Horizon values”

• The text devotes some attention to the question of whether
project analysis should try to estimate what happens at the
end (time T) of projects with a finite life.
• In certain cases, physical assets may have a “scrap value”,
which should be considered in the NPV calculation (as a
benefit at time T). In other cases, one should allow for “clean-
up costs” (a cost at time T) if it costs more do tear down a
physical asset than one can earn from selling parts of it for
scrap, or if natural sites have to be cleaned up and restored
• In other cases, at time T is arbitrarily chosen but an informal
estimate is made of any remaining value of the project
beyond that time. This is what the text refers to as a “horizon
value
– For most long-lived projects, these items may not be very important
Projects with different time horizons

• The text considers the case of two alternative energy
generation projects, a hydroelectric dam (H) that lasts 75
years and a “co-generation” project (C) that lasts 15 years.
• In the example, it is assumed that the discount rate is 8% and
that the NPV(H)=30, NPV(C)=24. Does that mean that H
should be chosen?
• Answer: no, because the time horizons are different
• One way to do a proper comparison is by assuming that if C is
chosen, another C project will be done at the end the 15 year
period of its useful life, then another one after 30 years,
another after 45 years, etc.
• Another way is to compute annualized benefits for the two
projects
• In either case, the C project will be the one that will be chosen
Annuities

• Annuity calculations are made when evaluating the PV of a
constant stream of payments (B or C) that lasts for a specified
time (say, t years). If the payment is A per year at the end of
each year (with the first payment coming a year from “the
present”, that is, from time 0), the PV formula is

1  (1  i )  n
PV ( A, i )  A 
where i is the discount rate.i
• The text notes that this is the formula for an ordinary annuity.
There is also something called an annuity due in which the
first payment A occurs now. For that type of annuity, the
formula is just like the one above, except that one adds
another A.
Annuities, cont’d

• Note also that if you want to compute the PV of a stream of
monthly payments, and interest is compounded monthly, you
use the same formula, except t is measured in months, and
the annual interest figure is divided by 12 (for comparability,
of course, A has to be divided by 12 as well).
• Note finally that if the benefit stream is constant at the annual
rate of A, and interest is compounded instantaneously, the
relevant PV formula is

T
PV ( A, i)   Ae dt  1  e
t 0
A
r
 rt   rT
          
Discounting and inflation

• Suppose that the discount rate that is being used is “the”
interest rate observed in the market (say, it is 8%). However,
suppose also that expected price inflation is 5% per year
• Suppose also that in estimating benefits and costs in the NPV
calculation, inflation has been taken into account. Therefore,
B and C in each year are given by

Bt  BCt (1.05 ) , Ct  CCt (1.05 )
t                     t

where the subscript Ct denotes what the Benefit or Cost would
have been in year t if there had been no general price
inflation (that is, what they would have been “at constant
prices”)
Discounting and inflation, cont’d

• The following two formulae can now be shown to be
equivalent:

( Bt  Ct )   ( BCt  CCt )
t 0 1.08 t t 0 (1  r )t
T             T

provided r is computed as         0.08  .05    i p
r            
(1.05 )    (1  p )
where i is the nominal (observed) interest rate, and p is the
expected rate of inflation
• Note that the text also discusses the case where B and C grow
at constant rates for other reasons. The calculations are
similar
Inflation, cont’d

• In words, this says that if there is expected to be general price
inflation in the economy, one can either compute the NPV by
estimating B and C flows taking inflation into account and
discounting by the nominal rate of interest (i in the example),
or estimate B and C “net of inflation” (with the subscripts Ct in
the previous notation), and discounting by the real rate of
interest (essentially, the nominal rate of interest minus the
expected rate of inflation)
• Most cost-benefit analysis is done using the second method,
as it means that one doesn’t have to forecast future inflation
rates when estimating B and C.
• However, future inflation rates must still be taken into account
when computing r, the real rate of interest
Measuring inflation accurately

• Estimates of expected inflation will be based on published
inflation data. Therefore, the real discount rate will only be
accurately measured if the inflation measurements are
accurate
• The reason why it is very important to have an accurately
established real discount rate is that resource allocation
decisions for long-lived projects will be very sensitive to the
discount rate, and the discount rate used in evaluation of such
projects will have an impact on the allocation of resources
between present and future generations of citizens
• So it is important to have procedures that yield accurate
estimates of inflation; later we discuss the Journal of
Economic Perspectives paper by Moulton (1996) on this
Some notes on Pindyck-Rubinfeld Ch 15

• The first four sections of PR covers discounting/NPV; it may be
useful for you to review it as an alternative exposition of the
same material as in the CBA text
• In CBA we usually only use a single interest rate as the
discount factor. In real capital markets, we see many different
interest rates and rates of return. In part the different rates
reflect differences in risk (section 15.5)
• Make sure you know what is meant by diversifiable and non-
diversifiable risk.
• A risk-free rate of return would be something like the rate of
return on a government bond
– but note that even a government bond carries an implicit risk: the
value of the payments may be reduced by inflation
Some notes on Pindyck-Rubinfeld Ch 15, cont’d

• In some countries, governments now issue indexed bonds.
These are bonds that pay an interest rate equal to the
observed rate of general price inflation plus a fixed
percentage return
• If such a bond is available, the fixed part of its return is a close
approximation to the risk-less inflation adjusted rate of
interest
• The rate of return on an asset such as a broad collection of
stocks will typically be higher than the risk-less rate of interest
– one way to earn that return is to invest in a mutual fund
– a mutual fund: a fund that owns a shares (stock) in many companies;
individual in turn buy a share in the fund
Some notes on Pindyck-Rubinfeld Ch 15, cont’d

• A broad-based mutual fund will have a high degree of
diversification because it owns shares in many companies, so
the variance in its rate of return will be smaller than that for
shares in an individual firm
• However, some economic shocks will affect the return on the
share in all companies
• The risk that stems from such economy-wide shocks cannot
be diversified away. Hence in equilibrium, the return on
capital in the stock market has a higher risk than that on a
risk-less government bond
• In the text, the risk-free return is denoted by rf while the
diversified stock market return is denoted by rm
– f and m are subscripts
Some notes on Pindyck-Rubinfeld Ch 15, cont’d

• Assuming savers (asset-owning households) are risk-averse to
some extent, they face a tradeoff between risk and return
– one way they can reduce their risk is to hold part of their total assets
in safe assets (government bonds)
• But they can also reduce their risk by trying to find companies
(or collections of companies) that have a tendency to do
relatively well when the market as a whole does badly
• Whether or not the shares of a particular company has this
characteristic depends on a parameter that measures
depends on the correlation between an individual company’s
profit and the return of the market as a whole
P-R Ch 15 and the CAPM

• This parameter is usually called β. If β=1, the return on this
company’s shares is correlated with the return in the market
as a whole and has about the same magnitude
• If β>1, this company’s shares have relatively volatile returns
that are positively correlated with the return in the market as
a whole
• If β is low (or even negative?), the return on shares in this
company does not move together with the average return in
the market as a whole
• Letting the equilibrium return on the shares in company i be
denoted by ri, the basic equation in the CAPM approach is
given by
ri  rf   (rm  rf )
P-R Ch 15 and the CAPM, cont’d

• So if we know β for a company and have estimates of rf and
rm, we can estimate the equilibrium return ri that owners of
the shares in this company will earn over time
• So owners of the shares in a company with a high β will earn a
high rate of return on average over time
– but the price they pay is accepting a higher amount of non-
diversifiable risk
• PR now state that if a given company wants to use the NPV
criterion to evaluate the profitability of an investment that
will have the same β as the company’s other assets, then the
discount rate it should use should be ri.
– that is, it should only go ahead with a project if it has a positive NPV
using the discount rate ri
P-R Ch 15 and the CAPM

• The logic is as follows. If an investment project has an NPV<0
at the discount rate ri, that means that the rate of return on
the investment is <ri.
• But since investors will only hold shares in a company with
this degree of riskiness if they earn, on average, a return of ri,
if the company invests in a project that yields less than ri, the
value of the company’s shares will fall
• Since the task of management is to produce the maximum
value for shareholders, they should not invest in a project that
has an NPV<0 when discounted at ri
Relevance to CBA

• The CAPM model suggests how the discount rate for a private
firm should be determined
• Later we will discuss how the discount rate for social Cost-
Benefit Analysis should be chosen; does the CAPM model
have any relevance to this issue?
• Answer: yes, in principle it does. If there is uncertainty about
what return a project will yield, and if one can estimate to
what extent the future return is correlated with non-
diversifiable risk in the economy (that is, the project’s β), then
projects with higher β should be discounted at a higher rate
than those with a low β
P-R Ch 15: investments by consumers; human capital

• The PR text also has a few pages discussing consumers’
investments in durable goods (like air conditioners, washing
machines, cars ..)
• These things cost money when you buy them (money you
could have saved and earned interest on), but will yield a flow
of benefits in the future
• So you will only buy them if your relevant discount rate is low
enough (why?)
• Evidence on the consumers’ implicit discount rate can be
obtained by looking at their choices among more or less
energy-efficient appliances
• The text cites research that suggests that consumers’ implicit
discount rates are very high
P-R Ch 15: investments by consumers; human capital

• An interesting finding is that the implicit discount rate is
higher for people with low income
– that is, high-income people are more likely to buy more expensive, but
more energy-efficient, appliances
• As the text notes, this may be because people with low
income are in debt (for example, they may owe money on
credit cards)
– what is your opportunity cost of funds if you owe money on a credit
card?
• So the logic here is: for a poor person, it makes more sense to
buy a cheap appliance (even if it less energy efficient),
because his rate of return on every dollar he uses to pay back
part of his credit card debt is so high
P-R Ch 15: investments by consumers; human capital

• As you know, an important investment for many people
consists in becoming educated
• It is an investment because it involves a cost early on, in
return for a return (partly in the form of higher income) later
– the cost is partly out-of-pocket (books, school fees), but mostly an
opportunity cost (in the form of not earning a wage from working
while you are going to school)
• The discount rate that you use (perhaps implicitly) in
evaluating this investment may determine how much
education you decide to get
– for example, many Masters’ students decide not to go on to the PhD;
why? Does the discount rate matter?
P-R Ch 15: optimum use of a depletable resource (oil)

• Suppose you are the owner of a quantity of oil in the ground.
It costs an amount MC per barrel of oil to pump it out and sell
it. The price of oil today is P>MC. You expect the price of oil to
rise in the future. What should you do?
– the answer depends on how fast you expect the price to rise in the
future, compared to your cost of capital (the interest rate)
• Discuss the logic: if you pump it all out and sell it today, you
can invest the money and earn an interest rate r. What if you
leave it in the ground?
– as the text shows, the logic of this model leads to the conclusion that
in long-run equilibrium, the price of a depletable resource should rise
over time at the rate of interest (why?)
– over time, less and less will be used per year
A digression: Åkerman’s problem

• When you plant a tree, the amount of wood in the tree is very
small
• As the tree grow over time, the amount of wood it contains
becomes larger and larger, until it reaches a maximum as the
tree matures
• Later as the tree becomes very old, the amount of wood in it
declines to zero, as the tree dies and the wood is broken
down through natural processes
• Question: if the price of wood remains constant over time, at
what age should we cut and sell the tree (and perhaps plant a
new one)? (Assume it costs 0 to cut it)
– this is called Åkerman’s problem because a Swedish economist by that
name first raised it in the literature
Åkerman’s problem
Quantity of wood                                       slope r

Q of wood

A

B

The tree should be
cut at age A*
(harvesting it at B*
would be too early)
Age of tree
(years)

B*          A*
How is the rate of interest determined?

• Answer: in the market for “loanable funds”
• In that market, individuals who want to save some of their
income will offer to lend their money
– the amount they want to lend may depend on the rate of interest they
can earn (net of risk): if the interest is high, they are likely to want to
lend a larger amount of funds
– (query: is this necessarily true? May there be an income effect that
offsets the substitution effect?)
• Who want to borrow money in this market? Answer 1: some
families or individuals who have expenditures that exceed
their income (for example, students)
• Answer 2: business firms that want to raise funds to invest in
productive capital (buildings, inventory …)
Supply, demand for loanable funds
r

S

r*

Tot D

B

C
Loanable
funds

L*
Supply, demand for loanable funds

• In this diagram, the curve S depicts the amount per year that
is offered in this market by families that save (spend less than
their income)
– many of those in this category are families with adult workers in their
prime earning years saving for retirement
• The curve C depicts the amount being demanded in this
market by families (for example, young families or students)
who need to borrow because they don’t have enough assets
to fund their current spending
• Curve B is demand for funds from business firms that plan to
invest in new capital for production purposes
• The equilibrium interest rate r* is where the supply of
loanable funds equals the total demand (from Business and
borrowing Consumers)
JEcP inflation paper

• As noted above, observed market rates of interest may need
to be corrected for price inflation in finding the appropriate
CBA discount rate. This raises the question:
• Do we know exactly what we mean by an “accurate” measure
of price inflation?
– Answer: yes, at the individual level at least. Specifically, an accurate
inflation measure should be based on the expenditure function: it
should measure the percentage change in the amount of expenditure
that would be necessary to keep an individual at the same level of
utility as at the earlier prices
• In real life, inflation measures are based on changes in fixed-
weight price indexes that are computed on the basis of
observed prices (“Laspeyre’s indexes”)
Inflation paper, contd

• In the paper, look carefully at the description how US
statisticians choose what prices to observe, in what cities, for
what varieties of goods, and in what types of sales outlets
• Also note that table provided of the expenditure shares in the
typical consumer budget; what would these budget shares
look like in China?
• The paper explains that the prevailing view is that inflation
measures based on the consumer price index in the US have
led to inflation estimates that are too high (by as much as 0.5-
1 % per year; some estimates are even as high as 1.5%)
JEcP paper on inflation

• The paper identifies five reasons why such a bias may arise:
the substitution bias, the formula bias, the outlet substitution
bias, the quality adjustment bias, and the new commodities
bias.
• From the viewpoint of economic theory, the most interesting
one is the substitution bias; note in this context the possible
solution using “superlative” indexes such as the Fisher index
or the Törnqvist index
• In practice, one suspects that the most significant reasons for
bias may be those relating to quality change, and new goods
JEcP paper on inflation

• In the context of quality change, an important technique has
been “hedonic regression techniques”, where the prices of
different varieties of a good have been statistically related to
each variety’s “characteristics”. These estimates can then be
incorporated to distinguish between price increases that are
due to quality changes and those due to general inflation
• Note also the interesting comment that if there had been no
adjustment for quality, the estimated price increase for the
passenger care component in the US Consumer Price Index
would have been 80% higher during 1967 to 1994.

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