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

suppose that the cost is covered by increasing some tax. 7 For the purpose at
hand, let us, however, assume that the funds are raised through the tax t2 levied
on good x2. The revenue collected per unit increase in this tax rate is denoted
R= t2 and the …nance requirement is that R = C, i.e., the extra revenue
R covers the monetary cost C. Faced with this tax increase, consumer h will
reduce his demand for good x2 by
x h 2 = @xh 2
@t2
t2
(35)
where @xh
2
@t2 is the slope of his demand function for good x2. We can use that to
rewrite the cost di¤erential ( SW F ) as follows:
SW F =
4X
h=1
@Uh
@xh @x
2
h 2
@t2
t2: (36)
From the de…nition of the …nance requirement, we can write t2 = C
R= t2 and
substitute that into the cost di¤erential:
The term @Uh
@x h 2
SW F =
C
4X
@Uh
R= t2 @x
h=1
h 2
@x h 2
:
@t2
@x h
2 represents, in units of utils, the welfare cost for consumer h
@t2
of the necessary tax increase. We can de…ne the associated welfare cost in units
of income by dividing through with the private marginal cost of income h, i.e.,
Uh
t2
1
h
@Uh
@x h 2
@x h 2
;
@t2
Uh where represents the welfare cost of the tax for consumer h in units of
t2
private income. Now, let us substitute this into the cost di¤erential:
SW F =
C
4X
R= t2
h=1
The term P4 h=1 h Uh represents the social (utility cost) of the tax increase
t2
needed to …nance the project and we could write this more compactly be denoting
it simply by SW F= t2 = P4 h=1 h Uh . This allows us to write the cost
t2
di¤erential as
SW F =
h
Uh
t2
:
SW F= t2
C: (37)
R= t2
7 It could be many taxes, in fact, if the tax-expenditure system were designed optimally
and in a joint-up way, then we know from the discussion of the Ramsey tax problem, that it
would be optimal to adjust all taxes in order to raise the funds at the lowest possible social
cost.
15