Transcript declining discount rate model

Chapter 8: Choosing a
Discount Rate
The Ideal Market for Loans
Demand for loans summarizes borrowers’ choices
Supply of loans summarizes lenders’ choices
The Supply of Savings and
Individual Time Preference
 Given the market interest rate, an individual
chooses how much of her current income to
consume immediately and how much to
save.
 Income which is saved collects interest,
leading to increased consumption in the
future.

Jill has $100 which she can spend now or
save for next year at 5% interest.
 If Jill consumes $50 and saves $50 this year
she will have $50•(1.05) or $52.50 next year.
The Efficiency of Capital and
the Demand for Loans
 The demand for loans in Figure 8-1 is based primarily on the investment
decisions of businesses.
 Rational and informed firms will choose to invest whenever the returns to
capital investment, corrected for factors such as added risk or inflation, are
greater than the returns to saving.
Figure 8-4:
A Savings-Investment Equilibrium
 Figure 8-4 the bulging line is a production
possibilities frontier
 The curve labeled U1 represents the highest
possible social indifference curve that can
be achieved given people’s tastes and the
limited productive resources of society.
 The straight line T represents the market price
of trading current for future consumption. In
this perfectly competitive case slope of this
line will be -(1+r*), where r* is the competitive
social discount rate.
The Effect of Taxes
 In Figure 8-5 both the supply of loans
(individual savings) and the demand
(returns to investment) are taxed.
 At investment level I1 investors will
receive a before-tax return of r3 and an
after-tax return of r2.
 Similarly, savings brings a before- tax
return of r2 but an after-tax return of
only r1.
 r1 is the social rate of discount because
it represents the marginal benefit to a
dollar of saving to individuals after
taxes. However, r1 is below the optimal
social rate of discount r* because of
the tax effect.
Welfare Loss from Taxes
 In Figure 8-6 the equilibrium occurs
at point A‘. At this point the slope
of the PPF equals –(1+r3) while the
slopes of budget line T and the
social indifference curve equal –
(1+r1).
 Also, at point A’ there is more
current consumption and less
investment and saving than at point
A.
 The lower social indifference curve
clearly shows the loss of efficiency
from this inequality in the marginal
rates of return
The Shadow Price of Capital Method
Estimating the Present Value of a project using the shadow price of capital
method takes 4 steps:
 Step One: Estimate the Project’s Effect on Investment
According to Harberger, nearly all borrowed funds can be assumed to displace
investment. On the other hand, projects funded through taxes are more likely to
displace consumption.
The Shadow Price of Capital Method
 Step 2: Annualize the Capital Cost
 The annualized value of capital cost is found by solving the following
present value equation for X
8-1) PVk 
X
X
X


...

1  r 3 (1  r 3) 2
(1  r 3) n
where r3 is the rate of return to capital and n is the number of years in the
lifespan of the displaced capital.
 This equation can be converted to a more useful form that is similar to the
annuity value equation introduced in Chapter 7.
r 3(1  r 3) n
(8-2) X  PVk
(1  r 3) n  1
The Shadow Price of Capital Method
 Step 3: Include these annualized capital costs from
equation (8-2) as a negative value in the net benefits of
the project.
 Step 4: Discount the stream of net benefits at the
consumption rate of interest (or social rate of discount).
The Shadow Price of Capital Method:
An Example for the Student
 Assume that the U.S. government is funding a remedial education and
employment program called “No Adult Left Behind.” The program is
funded through borrowing. The project requires an initial investment of $20
billion dollars, will bring benefits of $6 billion per year to society, and will
face operating costs of $2 billion per year after the initial training period in
year zero. Assume that displaced consumption and investment both equal
$10 billion dollars.
 Your Turn 8-1: If the project lasts 5 years, find the annualized lost
consumption from the 10 billion dollars of displaced investment using a
return to capital of 7 percent.
 Your Turn 8-2: Including the displaced consumption and the other costs
and benefits above, find the present value of the project lasting 5 years.
The Weighted Discount Rate
 An easier but less accurate method of dealing with the consumption versus
investment issue involves discounting the project using a weighted average
of the rate of return on capital and social discount rate on consumption.

(8-4) Weighted r = a•r1 + (1-a)•r3
where a is the fraction of the project’s cost which is financed by displaced
consumption and (1-a) is the fraction which displaces investment.
 With foreign lending included, the weighted discount rate becomes
(8-5) Weighted r with foreign lending = a•r1 + b•rf + (1-a-b)•r3
where r1 and r3 are the social discount rate and return to capital, rf is the interest
rate on foreign lending, and b is the fraction of the project’s cost financed by
foreign lending.
Other Issues in the Discount Rate
Inflation and the Discount Rate
 Concept: The real interest rate is the rate of interest corrected for inflation.
It is generally defined as the nominal rate of interest minus the inflation rate.
If the real interest rate is r, the nominal interest rate is i, and the inflation rate
is p, the approximate formula for the real interest rate is r = i - p.
 If you are using real costs and benefits, a real interest rate is correct. If your
costs and benefits are nominal, your discount rate should also be nominal.
 If you are estimating the present value of future net benefits, either adjust
them upward for future inflation or use a real interest rate when calculating
present value.
Inflation Bias in the Official C.P.I
 The most commonly used measure of inflation in the U.S. is the Consumer Price
Index for all urban workers (CPI-U). This index measures changes in the cost of a
fixed set of goods over time.
 A price index that measures the cost of a fixed set of goods is likely to
exaggerate the costs of inflation for a number of reasons.
 Substitution bias refers to the tendency of consumers to substitute away from
goods whose prices rise more than the average.
 Same store bias: The CPI ignores the gradual movement of consumption to
discount stores,
 New Product Bias: New goods tend to experience steep price decreases in
early years,
 Quality bias: Some goods increase greatly in quality over time, particularly
technological products such as consumer electronics and automobiles.
 In the 1990s the Boskin Commission released a detailed study of these biases
and concluded that the consumer price index was biased upward by about 1.1
percentage points per year.
“Chain Weighted” Price Indexes
 Chain-weighted price indexes are based on annual spending data rather
than a fixed market basket and therefore do not suffer from substitution,
same store, or new product bias.
 U.S. chain weighted indexes include the chain weighted CPI published by
the Bureau of Labor Statistics and the chain weighted GDP price index
published by the Bureau of Economic analysis, and the GDP price deflator,
which is similar to the GDP price index.
Alternative Price Index Values
 Note that the CPI-U is higher than the chain weighted indexes in nearly
every case.
Table 8-1: Alternative Inflation Measures*
INFLATION
MEASURE
(PERCENT)
2001
2002
2003
2005
2012
CPI-U
CPI-U
(CHAIN
WEIGHTED)
IMPLICIT
GDP PRICE
DEFLATOR
2.8
1.6
2.3
3.39
2.07
2.3
1.2
2.1
2.95
1.95
2.4
1.7
1.8
3.21
1.75
*For the GDP Implicit Price Deflator search the following Bureau of Economic Analysis website:
http://www.bea.gov/index.htm . For CPI inflation data, see the following Bureau of Labor Statistics
website, http://www.bls.gov/cpi/home.htm .
The Project’s Time Frame
 Interest rates vary with the length of time of the loan, and vary over time
with variations in demand and supply conditions.
 Therefore, the discount rate should be based on an interest rate for low risk
assets such as government bonds that have a time frame similar to that of
the investment project.
Intergenerational Equity and
Discounting for Long Term Projects
High Discount Rates and
Distant Future Present Values
 Policies related to long term problems such as nuclear waste or climate
change have effects lasting many generations into the future.
 Even 100 years is enough for a high discount rate to reduce present value
to near zero.
Table 8-2: Present Value in Different Agencies: 1980s
YEAR CBO (2%) OMB (10%)
Now
$1
$1
1
$.98
$.91
2
$.96
$.83
10
$.82
$.39
100
$.14
$.00007
2 Methods of Including Future
Generations in Present Value
 The utilitarian social welfare function, which weighs current and future
utilities equally, may allow some discounting based on growing average
incomes and declining marginal utility of income.
 The declining discount rate model which is commonly (but less clearly)
referred to as a hyperbolic discounting model, lowers the effective rate of
discount in future years.

The utilitarian social welfare function for different generations is:
(8-7) SWFU = U (B1-C1) + U (B2-C2) + U (B3-C3), where U refers to utility.
 Note that the utilities of the 3 generations are equally weighted and no discounting
occurs for utility.
 If future generations are richer and the marginal utility of income falls as income
rises, some discounting based on the falling marginal utility of income is
acceptable according to utilitarianism.
The Ramsey Formula and Discounting
 Given the utilitarian assumption that all generations’ utilities will be equally
weighted, the discount rate for income or dollar net benefits across generations takes
the following form:

(8-9)
rg = ηρg.
 where rg is the rate of discount across generations, η = marginal utility of
consumption, and ρg is the annual growth rate of consumption.
The Range of Discount Rates using the
Ramsey Formula
 Representative estimates of the marginal utility of consumption (η) include 1.6 and
0.7.
 The annual growth rate of consumption (ρg) is somewhere between 1.5 and 2
percent per year, the mean real growth rate in developed countries over the 20th
century. Therefore estimates for ηρg range from 1.05 to 3.2 based on these
representative figures.
 These rates are lower than most short term discount rates based on the shadow
price of capital or other methods.
Declining Discount Rates

Declining Discount rate models come in many forms, and have multiple justifications. For
example, one could define the present value of $1 in year t as

(8-10) PVddr = 1/(1+αt), where α is the discount rate and t is time.
 Table 8-3 compares the declining discount rate and standard present value using a 5%
rate.
PVddr
PV
Table 8-3: Declining Versus Fixed Discount Rates
YEAR 1 YEAR 4 YEAR 10 YEAR 50 YEAR 100
1
.95
.83
.67
.29
.17
(1+.05t)
1
.95
.82
.61
.09
.008
t
(1+.05)
 Table 8-3 compares values for the declining discount rate and standard present value
discounting formulas using a 5 percent rate of interest. This example demonstrates that
declining discount rate models are accurately named,
Conclusion
 The goal of this chapter was to provide a more detailed view of the
theoretical challenges in determining an ideal discount rate for public
projects.
 There is no generally accepted discount rate for public projects, although
economists have moved closer to a consensus over the past two decades
for relatively short run projects.
 The issue of intergenerational equity in very long run policy analysis remains
a controversial issue, however.