Benefit-Cost Analysis Course':
Distributional Weighting
BENEFIT-COST ANALYSIS
Financial and Economic
Appraisal using Spreadsheets
Ch. 11: Distributional Weighting
© Harry Campbell & Richard Brown
School of Economics
The University of Queensland
Accounting for Income Distribution
¾ In the preceding chapters we assumed that
each $ of net benefits of same value irrespective
of sub-referent group
¾ We need to acknowledge and account for:
9Government’s income distribution objectives
9 project’s distributional impacts
Accounting for Income Distribution
Atemporal vs. Intertemporal Distribution
¾ Atemporal – how income (and income
changes) distributed among individuals or
groups at present; ie. within present generation
Accounting for Income Distribution
Interpersonal distribution
¾ How income is distributed among individuals
or households
¾ See Tables 11.1 and 11.2
¾ Intertemporal – how income (and income
changes) distributed over time; ie. between
present and future generations
Accounting for Income Distribution
Measuring Interpersonal distribution
Accounting for Income Distribution
Measuring Interpersonal distribution
Table 11.2 Income Distribution by Deciles
Table 11.1 Distribution of Households by Annual Income
Annual Income ( Y = $189)
Less than $100
$100 to 200
$200 to 300
$300 to 400
$400 to 500
More than $500
% Households
30.7
42.9
13.4
5.8
2.7
4.5
© Harry Campbell and Richard Brown,
School of Economics, University of
Queensland, 2003
Households
Top 1 per cent
Top 2 per cent
Top 10 per cent
2nd decile
3rd decile
4th decile
5th decile
6th decile
7th decile
8th decile
9th decile
Bottom decile
% of Income
8.31
22.81
33.73
15.49
11.61
9.22
8.12
7.32
6.47
2.94
2.67
2.45
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Benefit-Cost Analysis Course':
Distributional Weighting
Accounting for Income Distribution
Measuring Inequality
Measuring Inequality
Income is obviously not equally
distributed – some individuals or groups
earn more than others
The degree of income inequality can vary
considerably between countries and, within
a country, over time.
Economists have devised ways of
measuring degree of inequality
Measuring Inequality
Measure
area
outside
the main
triangle
Cumulative % pop
0.2
Cumulative % income
Total
=(0.08x2)+0.04
=0.2
0.8
Gini coeff.
= (0.5-0.2)/0.5
Square =
0.2x0.2
=0.04
= 0.3/0.5 = 0.6
Changing Income Distribution
¾While one investment option could be superior to another from
an economic efficiency viewpoint, it might be inferior to the other
from a distributional perspective
¾The final choice among projects will depend on the relative
importance that the policy-makers attach to the economic
efficiency objective versus the income distribution objective.
¾It is to the explicit incorporation of distributional objectives in
benefit-cost analysis that we now turn.
© Harry Campbell and Richard Brown,
School of Economics, University of
Queensland, 2003
If income was
equally
distributed the
Lorenz curve
would lie on
the diagonal
Figure 11.1 The Lorenz Curve
100
Cumulative %
population
The flatter the Lcurve, the smaller
the shaded area,
the less the degree
of inequality
0
0
100
Cumulative % income
GINI
Coefficient =
shaded area
as a % of
whole triangle
Changing Income Distribution
Assume 20% population earns 80% of income
Each triangle
= 0.8x0.2)/2
= 0.08
Accounting for Income Distribution
¾government can affect the distribution of income
between sectors and regions of the economy through
the sectoral and regional spread of public investment
projects
¾the type of investment undertaken can influence the
distribution of income among various categories of
income-earners
¾an investment that uses relatively more labour than
capital will imply more employment, and perhaps, a
larger share of income for the wage-earner versus the
profit-earner, and vice versa
¾different types of investment will have different
implications for the employment (and income) of
different types of labour
Distributional Weighting
¾Suppose that we are required to advise on the best choice of
projects taking into consideration the government's commitment to
the twin objectives of economic efficiency and improving income
distribution
¾In Table 11.5 we have before us 3 possible projects, A, B and
C, of which only one can be undertaken.
¾Each project affects income distribution differently
¾How can we incorporate government’s income distribution
objective into project choice explicitly?
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Benefit-Cost Analysis Course':
Distributional Weighting
Distributional Weighting
Distributional Weighting
What about choosing between D and E?
Table 11.5 Comparing projects with different atemporal distributions
Table 11.6 Comparing projects with different aggregate benefits and distributions
Referent Group Net Benefits ($NPV)
Project
Rich
Poor
Total
A
B
C
60
50
20
40
30
80
100
80
100
Referent Group Net Benefits ($NPV)
¾Project B can be rejected purely on economic efficiency
grounds - its aggregate net referent group benefits are less than
those of A and B, and the distribution of benefits among the rich
and poor is less egalitarian than that of either A or C.
¾the question is whether to choose A or C?
Project
Rich
Poor
Total
D
E
60
40
40
50
100
90
¾ Project D would be preferred on purely economic efficiency
grounds, whereas Project E might be preferred on purely
distribution grounds.
¾As long as there is a commitment to the objectives of
economic efficiency and income distribution, a conflict arises.
Choose D and we sacrifice distribution; choose E and we
sacrifice efficiency
Distributional Weighting
Distributional Weighting
¾This choice is a classic example of what
economists call a trade-off.
¾ If the government did not attach different
distributional weights to the net benefits accruing
to different groups, projects would be selected
purely on the basis of their aggregate referent
group net benefits.
¾assume we weight each additional dollar of net
benefit received by the poor by three times as much
as each additional dollar received by the rich
¾ What would this imply about the government's
objective concerning income redistribution?
Table 11.7 Applying distributional weights to project net benefits
Referent Group Net Benefits
($NPV)
Project
Rich
Poor
Total
D
E
60
40
40
50
100
90
Weighted (Social) Benefits
($)
¾ From what we have seen, it could mean one of
two things: either
Rich
(a) it does not regard project selection as an
important means of redistributing income; or,
Poor
(60x1.0)+(40x3.0) = 180
(40x1.0)+(50x3.0) = 190
Deriving Distributional Weights
Theoretical Basis
(b) it does not care about income distribution; i.e. it
attaches equal weight (1.0) to all sub-referent
group members.
Deriving Distributional Weights
Theoretical Basis
Diminishing marginal utility of consumption
Figure 11.2
Total
Diminishing marginal utility of consumption
F ig u re 11.3
Total U tility C urve
12
32
Utility
M a rg in al U tility C u rve
11
23
M arginal
Utility
9
12
1
2
3
Consumption level (units)
© Harry Campbell and Richard Brown,
School of Economics, University of
Queensland, 2003
1
2
3
Con sump tion level (u nits)
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Benefit-Cost Analysis Course':
Distributional Weighting
Deriving Distributional Weights
The appropriate income distribution
expressed in algebraic form as follows:
d
di
=
i
= (
Y
Y
)
weight
can
be
n
As an example of the use of such weights, suppose that a
net beneficiary of a project is in an income group which has
a level of income equal to, say, $750 per annum (Yi), and
that the national average income is $1500 (); the distribution
weight for that individual would then be:
1
the distribution weight for income group I
Y bar =
the average level of income for the economy
Yi
the average income level of group I
=
Deriving Distributional Weights
n
=
the elasticity (responsiveness) of marginal utility
with respect to an increase in income, expressed as the ratio
of the percentage fall in marginal utility to the percentage rise
in income
Deriving Distributional Weights
Someone at an income level of $2500 per annum will have
his/her benefits weighted by a factor of 0.66
An additional $1.00 going, for example, to someone
earning an income of $4250 per annum would be
valued at only 43% of the value of an additional $1.00
going to someone at the average ($1500 per annum)
income level.
di =(
1500 n
)
750
= 2n
If n = 0.8, then an individual at consumption level
$750 per annum will have her benefits weighted by a
factor of 1.74
Deriving Distributional Weights
Figure 11.4
W eighting Factors for E xtra Incom e
Weight
1.74
1.00
0.66
0.43
750
1550
2250
4250
Income level ($)
Deriving Distributional Weights
The higher the value of n, the faster the rate at which
marginal utility falls.
Table 11.8 Responsiveness of distributional weights to changes in n
Distributional Weight d i
$/Annum
250
750
1250
1500
1750
2250
2750
3250
3750
4250
n=0
n=1
n=2
n=3
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
6.00
2.00
1.20
1.00
0.71
0.67
0.55
0.46
0.40
0.35
36.00
4.00
1.44
1.00
0.73
0.44
0.30
0.21
0.16
0.12
216.00
8.00
1.73
1.00
0.63
0.30
0.16
0.10
0.06
0.04
© Harry Campbell and Richard Brown,
School of Economics, University of
Queensland, 2003
Deriving Distributional Weights
The formula for di expresses the point that the distributional
weights we use are determined by two factors:
(a)
the relative income/consumption level of the project
beneficiaries ; and,
(b) the value-judgement that is made about the utility or
satisfaction that is gained by project beneficiaries of different
income levels; the value of n.
If we do not apply distributional weights explicitly, we are
implicitly assigning a value di = 1.00 to each and every project
beneficiary, irrespective of how much (s)he earns
In some circumstances we can infer from the choices the
decision-maker makes what (s)he regards as the appropriate
or “threshold” distributional weights, but, it does not provide
any independent information about the appropriate weights.
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Benefit-Cost Analysis Course':
Distributional Weighting
Using Distributional Weights
To apply distributional weights we would need to know the
following about the project in question:
(a) identification of the project's gainers and losers;
(b) classification of the project's gainers and losers; i.e. to
which particular income category they belong; and,
(c) quantification of gains and losses, i.e. by how much do
the net incomes of the gainers and losers increase or
decrease?
The relevant project level information about (a) and (c)
already exists, for this is, in effect, what is contained in the
Referent Group analysis
Using Distributional Weights
10%
15%
20%
NPV A
NPV B
$360
$315
$270
$200
$150
$100
A bottom-up approach to derivation of distributional weights
¾ We prefer to consider benefit-cost analysis and
distributional weighting from the perspective of it serving
as a potentially powerful means of identifying the possible
distributional implications of government project selection
¾ It thereby sharpens our understanding of the value
judgements implicit in government decisions.
¾ In other words, through benefit-cost analysis we can make
ourselves, as projects analysts, and the relevant policymakers more aware of the consequences of decisions
concerning project selection.
Using Distributional Weights
Table 11.10 Distributional Weighting in the NFG Project
Table 11.9 Threshold Distributional Weights
Discount Rate
Using Distributional Weights
Threshold Distributional
Weight
(NPV[A] = NPV[B])
1.8
2.1
2.7
If the decision-maker considers it reasonable to value net
benefits generated in the Southern Region (B) at roughly two
to three times as much as those generated in the Central
Region (A), (s)he would favour alternative B even if the
discount rate were as high as 20%.
Intertemporal Distribution
¾what is not consumed by us today, is saved
¾Savings finance investment, and investment today generates
consumable output in the future.
¾Therefore, the decision we make today, regarding how much
income is spent now on immediate consumption and how much is
saved now for future consumption, is a decision about how
consumption should be distributed among those living today and
those living in the future.
¾The more that is consumed today, the less that is left for future
generations, and vice versa.
¾If the social time preference rate is lower than the market rate,
the present value of the additional benefit that could be generated
by one extra dollar of savings is greater than that of the additional
benefit generated by one extra dollar of consumption.
© Harry Campbell and Richard Brown,
School of Economics, University of
Queensland, 2003
Government
Landowners
Wage Earners
Downstream Users
Aggregate
Net Benefit(5%)
($000s)
dI
-173.2
1.00
33.2
0.6
53.2
4.0
-115.1
0.5
-201.9
Weighted Net
Benefit ($000s)
-173.2
19.92
212.8
-57.55
1.97
With this set of distributional weights, the weighted value of
aggregate referent group net benefits is positive, indicating
that from a ‘social’ perspective, the project is worthwhile.
Intertemporal Distribution
¾The lower social discount rate makes no allowance for the
different effects of projects on saving and reinvestment of project
net benefits.
¾It then becomes important to establish what part of the referent
group’s net benefits are saved and what parts are consumed, with
a view to attaching a premium on that part which is saved
¾It cannot be assumed that all members of the referent group
save the same proportion of any income gained or lost
¾Once we introduce a premium (or shadow-price) on savings we
need to make an additional adjustment to the raw estimates of
referent group net benefits
¾It is possible that the decision-maker will be faced with a tradeoff between a better atemporal and a better inter-temporal
distribution of income. (See Example 11.2.)
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Benefit-Cost Analysis Course':
Distributional Weighting
Intertemporal Distribution
Example11.2: Incorporating
distributional effects
atemporal
and
inter-temporal
¾Suppose that the effect of two projects can be summarized as
follows:
Project A generates a net referent group benefit of $100. $80 is
saved and $20 is consumed by a group with above average
income. Assuming that $1.00 saved is worth the same to the
economy as $1.20 consumed (i.e. the shadow-price of savings is
1.2), and that the distributional weight for this group is 0.75 then:
Net Benefit (in terms of Dollars of consumption)
=
$80(1.2) + $20(0.75)
=
$96 + $15
=
$111
Intertemporal Distribution
Consider the hypothetical example in Table 11.12 in which
the consumption levels of the different income groups are
shown in the first column and the atemporal distributional
weights are given in the second column. In this example, the
mean level of consumption is $1500
T a b le 11 .12 C o m p o site D istrib u tio n al W eig h ts
C o n s u m p tio n $ /an n u m
( C i)
D is trib u tio n a l
W e ig h t
(d i)
25 0
75 0
6.00
2.00
1.20
1.00
0.71
0.67
0.55
C c = 12 5 0
C = 15 00
17 50
22 50
27 50
Project B also has a net benefit of $100. Of this $40 is saved and
$60 is consumed by members of the referent group who enjoy an
average level of income. As the same shadow price of saving
(1.2) applies, and the sub-group’s distributional weight will have a
value of 1.0, then:
Net Benefit (in terms of Dollars of consumption)
=
$40(1.2) + $60(1.0)
=
$48 + $60
=
$108
In this instance the combined effect of introducing both
distributional weights is to favour the project that benefits the
relatively richer group, Project A, whereas in the absence of a
shadow-price of saving Project B would have been favoured.
Example11.3: Deriving the premium on savings indirectly
C0 = critical
consumption
level
Intertemporal Distribution
© Harry Campbell and Richard Brown,
School of Economics, University of
Queensland, 2003
Implicit premium
on savings = 1.2
Intertemporal Distribution
¾If distribution objectives are to be accommodated by a system of
atemporal and inter-temporal weights it will be necessary to
disaggregate net benefit for each referent group gainer or loser
into its consumption and savings components, and then weight
the consumption component by the atemporal distributional weight
and the savings component by the inter-temporal distributional
weight (or savings premium).
¾If we were to follow this procedure in the context of the NFG
project it would be necessary to disaggregate each stakeholder
group’s net benefits into their consumption and savings
components, and then apply the respective di to the consumption
benefit and savings premium to the savings benefit of each
referent group beneficiary or loser
6