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

groups of individuals, say rich, middle income and poor; or men and women,
etc.). This is, perhaps, most clearly seen by rewriting the equation as
NB = P H
h=1
b
h x h 1
P H
h=1
c
h x h 2; (12)
where the welfare weight given to individual h in the calculation of the bene…ts
(superscript b is for bene…t) is
b @SW F @Uh
h = =
@Uh @x1
@SW F
@Uh
hq1: (13)
The weight given to individual h in the calculation of the opportunity cost
(superscript c is for cost) is:
c @SW F @Uh
h = =
@Uh @x2
@SW F
@Uh
hq2: (14)
The two weights might di¤er insofar as the marginal utility of the output delivered
by the project is di¤erent from the marginal utility of consumption foregone
as a consequence of the project. Nonetheless the principle is clear: we need to
de…ne a sensible set of welfare weights and then weight the costs and bene…ts
accordingly.
This makes it more complicated to estimate the shadow prices. Recall in the
simple world with one representative consumer, we de…ned the shadow price of
say the output x1 as the change in social welfare induced by a small change in
project output relative to the marginal value of private consumption, i.e.,
@SW F
@x1
1 = @SW F
@U
@U
@x1
1 : (15)
When there are many di¤erent consumers, the change is social welfare induced
by the project output is
HX @SW F
h=1
@Uh
@Uh
@x h 1
x h 1: (16)
In the special case, where all consumers bene…t by the same amount from the
project x h 1 = x1, the increase in social welfare induced by a small change
in the (common) project output is P H
PH h=1
h=1
@SW F
@Uh
@Uh
@x h 1
, which we see is equal to
b
h. But in the case of heterogenous consumers there is not a single
marginal value of private income: there are, in principle, H di¤erent ones; one
for each consumer. So how do we normalize the change in social value to get at
the shadow price? One possibility is to use the average value = 1 PH H h=1 h
and de…ne the shadow price of the (common) output as
q SP
1
=
P H
h=1
b
h
(17)
but we could also choose, for example, to use the marginal value of money for
a particular consumer, say, consumer 1, 1 as the unit of account.
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