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Discounting

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					Discounting

How should the future benefits of a
project be weighed against present
costs?
Generic Group Project

 You are making a recommendation
  about using catchment basins for
  groundwater recharge in LA. Costs
  now provide water in future, offsetting
  future water costs.
 Good idea?

 Big issue: comparing costs today with
  benefits tomorrow
    “Contractor wins
    $314.9 million Powerball”

 Winner opts for $170 million lump-sum
  payoff instead of 30 annual payments of
  about $10.5 million per year.
 Question: Why would someone choose
  $170 million over $315 million?
 Answer: The time value of money. Future
  earnings must be discounted.
Outline

 What is discounting?
 Why do we discount?

 The mechanics of discounting.

 The importance & controversy of
  discounting.
 Discounting in practice.
        What is discounting?

   Public and private decisions have
    consequences for future:
     Private: Farmer invests in water-saving
      irrigation. High up-front cost, benefits accrue
      over time.
     Public: Dam construction/decommissioning,
      Regulating emissions of greenhouse gases,
      wetlands restoration, etc.
   Need method for comparing costs & benefits
    over time.
Why do we discount?

 Put $100 in bank today, get about $103
  next year.
 Why does money earn positive interest?
     People generally prefer to consume
      sooner rather than later (impatience),
     If we invest, we can get more next year
      (Productivity of capital).
         Example: Carol’s Forest

   Assume forest grows at a declining annual rate
       Annually: 4%, 3.9%, 3.8%,….
   When should she cut her forest?
   If she’s patient: wait and get more wood
   If she’s impatient: cut now
   Tension: impatience to consume vs. waiting and
    producing more
   Interest rate is an “equilibrium” between
    impatience of consumers and productivity of the
    forest
Mechanics of discounting

 Money grows at rate r.
 Invest V0 at time 0: V1=V0(1+r)
 V2=V1(1+r),…
 Future Value Formula: Vt=V0(1+r)t.
 Present Value Formula: V0 = Vt/(1+r)t.
     This is compounded annually
     Continuous: V0 = Vtexp(-rt)
   Other formulae available in handout.
The drip irrigation problem

   Farmer has to decide whether to invest in
    drip irrigation system: should she?
   Basic Parameters of Problem:
       Cost = $120,000.
       Water savings = 1,000 Acre-feet per year,
        forever
       Water cost = $20 per acre foot.
   Calculate everything in present value
    (alternatively, could pick some future date
    and use future value formula)
       Investing in drip irrigation
       (r=.05)

Year       Costs     Benefits   Cumulative
                                Net Gain
0          120,000   20,000     -100,000

1          0         19,048     -80,952

2          0         18,141     -62,811

3          0         17,277     -45,534
      When does she break even?

                         Drip Irrigation Project

             200000
             150000
             100000
Net Payoff




              50000
                  0
              -50000 0      5       10          15   20   25

             -100000
             -150000
                                         Year
              Concept of Present Value
              (annual discount rate r)

   What is the present value of a stream of costs and benefits, xt: x0,
    x1,…,xT-1
   PV= x1 + (1+r)-1x2+(1+r)-2x2+…+(1+r)-(T-1)xT-1
   If PV > 0, stream is valuable
   Annuity: Opposite of present value – convert a lump-sum into a steam
    of annual payments
        Eg: spend $1,000,000 on a dam which is equivalent to $96,000 per
         year for 30 years (check it!)
        Eg: Reverse mortgages for seniors
        Where does inflation come
        in?
 Inflation is the increase in the cost of a
  “basket of goods” over time.
 Your grandpa always says “An ice cream
  cone only cost a nickel in my day”….the
  fact that it’s now $2 is inflation.
 Want to compare similar values across
  time by controlling for inflation
     Correct for inflation: “Real”
     Don’t correct for inflation: “Nominal”
The “Consumer Price Index”

 CPI is the way we account for inflation.
 CPIt = 100*(Ct/C0)
     Ct = cost of basket of goods in year t.
     C0 = cost of basket of goods in year 0.

   E.g. Year          CPI
        1990           100
        1991           104.2
        1992           107.4
Some other discounting concepts

 Net Present Value (NPV): The present
  value of a stream of values over the
  life of the project (e.g., NPV of B-C)
 Internal Rate of Return (IRR): The
  interest rate at which project would
  break even (NPV=0).
 Scrap Value: The value of capital at
  the end of the planning horizon.
Importance of discounting

   Discounting the future biases analysis
    toward present generation.
     If benefits accrue later, project less likely
     If costs accrue later, project more likely
     Speeds up resource extraction
     E.g., lower discount rate increases
      desirability of reducing GHG now (WHY?)
   “Risk-adjusted discount rate”
       Risky projects may justify increasing
        discount rate.
Social vs. private discount
rate
   Private discount rate easily observed
     It is the outcome of the market for
      money.
     Depends on risk of default on loan.
   Social rate may be lower
     People care about future generations
     Public projects pool risk – spread
      losses among all taxpayers.
     Argues for using “risk-free” rate of
      return.
Social discount rate in
practice
 Small increase in r can make or break
  a project.
 Typical discount rates for public
  projects range from 4% - 10%.
 Usually do “sensitivity analysis” to
  determine importance of discount rate
  assumptions.
 Be clear about your assumptions on r.
    Weitzman’s survey (2160
    Economists)
 “Taking all relevant considerations into
  account, what real interest rate do you
  think should be used to discount over
  time the benefits and costs of projects
  being proposed to mitigate the possible
  effects of global climate change?”
 Mean = 4%, Median = 3%, Mode = 2%
                                Discount Rate Choice


                 500


                 400


                 300
Responses




                 200


                 100


                   0
            -5          0   5        10        15      20   25   30

                 -100
                                     Discount Rate
Far-distant costs or benefits

   Many important environmental problems
    have costs and/or benefits that accrue far in
    the distant future.
   Constant-rate discounting has 3
    disadvantages in this case:
       Very sensitive to discount rate
       Far distant consequences have little or no
        impact on current policy
       Does not seem to fit empirical or
        experimental evidence very well
Constant-rate discounting
                                                NPV

                     1,200


                     1,000
 PV of 1000, T=100




                      800


                      600


                      400


                      200


                        0
                             0   0.02   0.04          0.06        0.08   0.1   0.12
                                               Discount Rate, r
New Innovations
   Uncertainty: If uncertain over future value of r, then
    “as if” rate was lower.
   Hyperbolic discounting: Gives much more weight to
    future:
      Formula: PV=FV/(1+at)g/a
      But is time-inconsistent: “If social decisionmakers
        were to use people's 1998 hyperbolic rates of time
        preferences, plans made in 1998 would not be
        followed - because the low discount rate applied to
        returns in, say, 2020, will become a high discount rate
        as the year 2020 approaches.” (Cropper and Laibson)
   Quasi-hyperbolic discounting can be time consistent
Hyperbolic vs. Const Rate

                                           a=1, g=.1, r=7%

                         1,200


                         1,000
 PV of 1000 in t years




                          800

                                                                          hyp
                          600
                                                                          reg

                          400


                          200


                            0
                                 0   20   40     60      80   100   120
                                                 t

				
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