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					          CBA 100401

Discounting, Inflation, and the Social
           Discount Rate
                         Guide to readings

• 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
                    Guide to readings, cont’d

• 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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