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

1. The project has the same impact on the consumption of good x1 and x2
for all consumers, i.e.,
x h 1 = x1 and x h 2 = x2 for all h:
2. All consumers have the same private marginal value of income, i.e., h =
for all h. This, in turn, is satis…ed if (for example) all consumers have the
same utility functions (tastes) and the same level of income. 1
3. The social welfare function values utility increments of all consumers in
the same way, i.e.,
@SW F
@Uh
= @SW F
@U
for all h:
This is, for example, satis…ed by the (unweighted) utilitarian social welfare
function where = 1 for all h.
@SW F
@Uh
Under these (su¢ cient) conditions, we can write equation (8) as
@SW F
NB = H
@U
[q1 x1 q2 x2] (9)
and that takes us back to the scenario considered in Lecture note 13-14. The
three conditions are su¢ cient, so we can get away with less. If, for example,
we got conditions 2 and 3, but not condition 1 (i.e., the project might deliver
di¤erent amounts to di¤erent individuals or cost them di¤erent amounts in
forgone consumption), we would get
NB =
@SW F
@U
PH [q1 h=1 xh PH 1 q2 h=1 xh2]; (10)
which e¤ectively allow us to add the bene…t and cost increments up and then
value them at market prices. Another possibility is that conditions 1 and 3
holds but not condition 2. This case we get that
NB =
@SW F
@U q1
PH x1 h=1 h
= [q1 x1 q2 x2]
@SW F
@U
@SW F
@U q2 x2
P H
h=1 h
P H
h=1 h: (11)
Whichever of these we take as the benchmark, it should be clear now that
we have to assume a lot for it to be valid to use the simply cost-bene…t rule
that the project will increase social welfare if the market value of its output is
greater than the market value of the inputs. 2 In fact, one could argue that these
1 Other assumptions can also insure that everybody have the same marginal value of income.
For example, if consumers have quasi-linear preferences, then they can have di¤erent
preferences over some goods and di¤erent levels of income while preserving a common value
of the marginal utility of income (typically assumed to be one).
2 There is, of course, also the issue of whether market values exist and whether they represent
the shadow value of the outputs and inputs.
4