Lecture Note 15: Social Cost Benefit Analysis - University of ...

of x2 by x h 2 for consumer h. Under what conditions, we can judge the social
value of this project without having to worry about distribution. The answer
to this question will give us a good sense of whether distribution is a central or
a marginal issue for social cost bene…t analysis.
By de…nition the net bene…t (NB) of the project is the di¤erence between
social welfare before and after, so we can write it as
NB = SW F (U1(x 1 1 + x 1 1; x 1 2 x 1 2); :::; UH(x H 1 + x H 1 ; x H 2 x H 2 ))(4)
This can be approximated by
SW F (U1(x 1 1; x 1 2); :::; UH(x H 1 ; x H 2 )):
NB = PH @SW F @Uh
h=1 @Uh @xh 1
x h 1
PH @SW F @Uh
h=1 @Uh @xh 2
x h 2: (5)
The …rst term on the right-hand side captures the social bene…t of the project.
For a particular consumer h, the extra consumption of good x1 is x h 1. The
extra utility of this is found by multiplying x h 1 by the marginal utility of
good x1 ( @Uh
@x h 1
). The social value of this increase in utility for consumer h is
then found by multiplying by the marginal increase in social welfare per unit
@SW F
of utility enjoyed by consumer h ( ). Finally, the total social bene…t is
@Uh
found by summing over all consumers. The interpretation of the second term
– representing the social opportunity cost – is similar, but you might just go
through the details on your own to make sure you understand what it is.
All consumers maximize their utility subject to given market prices and their
given income, so before the project is introduced, the quantities consumed by
each consumer h is determined by the following two conditions:
@Uh
@x h 1
@Uh
@x h 2
= hq1 (6)
= hq2; (7)
where h is the marginal utility of income of consumer h. We note two things
here. Consumers consume di¤erent amounts of the two goods partly because
they have di¤erent tastes (i.e., their utility functions are di¤erent) and partly
because they have di¤erent incomes (which contributes to making their private
marginal utility of income di¤erent). They might also face di¤erent market
prices, but we leave that complication aside here. We can substitute these
optimality conditions into equation (5) to get:
NB = PH @SW F
h=1 @Uh
hq1 x h 1
PH @SW F
h=1 @Uh
hq2 x h 2: (8)
Now, we are in a position to answer the question we set out to answer, i.e.,
when can we ignore the distributional impact of a project in our calculation of
its social value? It is useful to start by specifying some su¢ cient conditions:
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