# refers to the specific consumption group And secondly, v is designed to allow for the different values assigned to public income and

private sector consumption.

Substituting for (i) in equation (2.07):

NSB* = E - C (3 “ d/v) (2.13)

where C(3 - d/v) = distribution impact,

such that the distributional impact, the increase in private sector consumption, reflects both the cost of the reduction in public income measured in foreign exchange, 3, and the social benefit of additional

consumption in the private sector, d/v. Here, v reflects the public revenue constraint so that the higher the v or the scarcer the public income,

the greater the chances that projects will be selected which do not involve a significant transfer of resources from public to private sector consum­ ption. In other words, when v is greater than unity, public income

(or investment) is considered more valuable than average private

consumption. This implies that the government values future consumption to present consumption. A final note on NSB* is that the value d is intended to bias project selection in a manner that the private sector consumption which is generated by project investment will accrue

primarily to the poor. Thus "given the cost of the resource transfer,

3, the offsetting social benefit is determined in the light of the

overall constraint on public income, v, and the value of providing

2.1.4.3 Consumption Distribution Weight^ (d) The specification of a social welfare function precedes the

derivation of distribution weights. The LM/ST methodology recommends the use of an iso-elastic utility function. This function is specified as:

Uc = c"n (2.14)

where = marginal utility at consumption level c;

c = level of consumption; and

n = elasticity of marginal utility with respect to consumption.

Total utility, L'c , is obtained by integrating equation (2.14),

i.e. ,

U(c) = -i- C 1_n, if n > 1 (2.14b)

1-n

U(c) = log if n 1 (2.14c)

In accordance with the theory of diminishing marginal utility, the negative sign for n represents a falling value for the elasticity of marginal utility as income rises. Admittedly, there are other formulae that could be used to depict the diminishing nature of marginal utility. However, equation 2.14 is preferred because n evokes a significant

meaning: the higher the n, the more egalitarian the government's objective.

How the value of n is to be chosen is a matter of value judgement that is determined by policy objectives of the government. As in

traditional analytical methods, n is given a value of zero such that all

additional consumption is valued equally regardless of the recipients'

existing level of consumption.

The relevant value of d for any particular income recipient may change over time if his consumption level and average consumption

are growing at different rates. If, for instance, consumption substantially

rises from to then the increase in utility arising from this non­

marginal change in consumption relative to the marginal utility of -n

consumption at the average level of consumption c (the numeraire) is

U(c ) - U(c^). In terms of the numeraire, this is expressed as:

_ - n

(U(C ) - U(c1 ))/c

applying the weight, d, directly to (c^-c ) gives the normalized utility value: (C2_Cl)d = U(C2} " U(C1} -n c d U(c ) - U (c ) (- - 7- - - ) (c2 - c > ; such that 1 (2.15)

where d = consumption distribution weight for non-marginal

changes;

c^ = consumption without the project; and

c^= consumption with the project.

Depending on the values of n, d assumes the following forms:

1 For marginal change in consumption, the formula is

d = Uc/Uc = (c/c) n

where d = consumption distribution weight for marginal change; U-= marginal utility at average level of consumption, c.

, 1-n 1-n ;n (c2 ' c i ) d = -- ---- --- -— for n ^ 1; and (2.16) (1-n) (c 2 —c ) - (log c - log c ) d = --- -— --- -— =■ for n = 0 (2.17) °2 - °1

Here, the values of c and n are not project specific but country

specific. Being a value judgement, n makes d itself also a value judgement.

2.1.4.4 Summary Distribution Measure (D)

The effects of a project on consumption distribution are not always traceable or may affect all income classes. Where such cases persist, the methodology recommends the use of a 'global' distribution weight, D. This parameter is defined as 'the ratio of the social value of an additional unit of consumption distributed according to the present income distribution pattern to the social value of an additional unit of private consumption of a person at the average level of 'income' (Bruce, 1976, Annex A, p.l),D is not dependent on the income distribution impact of the project hence, it is categorized as a country parameter.

By definition, D will equal unity when n is equal to either zero or one. However, when the elasticity of marginal utility of income is

> 2

non-zero but greater or less than unity (0 < n < i) then,

D = 3n O - l ) X ~ n

(n + 3 - 1 )

where 3 (sigma) = distribution of consumption parameter.

(2.18)

1 See SVT (1975, pp. 66-67, 104, 137-139).

2 For the mathematical derivation of this equation see pp. 7-9 of

Setting 1 implies that there is a perfectly egalitarian distribution of consumption while allowing it to approach to zero means distribution of consumption becomes inegalitarian. How 3 is derived explicitly depends on the estimate of the Gini-coefficient of the income

distribution of a country,"*” viz:

GINI or (2.19)

1 + GINI

2 GINI (2.20)

2

2.1.4.5 Social Value of Public Income (v)

This parameter may be interpreted as a weighted average of the value of different types of public expenditure, the weights being the proportion of each in the marginal unit of expenditure. Its premise is based on the assumption that at the margin public sector income measured

in foreign exchange is used for different purposes such as education, defence, consumption subsidies, investment and so on. Thus v could be derived from:

v = £ a^ V j , and (2.21)

£ aj = 1 (2.22)

j