**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**