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

from an extra unit of income. How much lower depends on the curvature of
the function g, i.e., how fast marginal utility falls with income. It would seem
reasonable to say that a function g exhibits more inequality aversion if marginal
utility of income falls fast with income than when it falls slowly. After all, in
the later case, it does not really matter at the margin if there are big income
inequalities or not, while in the former case it does matter a great deal. The
speed at which marginal utility falls is controlled by the second derivative of the
function, i.e., by @2 g
@y 2 = ay a 1 . We could, of course, use the (absolute) size of
the derivative as our measure of inequality aversion, but such a measure would
not be invariant to a positive monotonic transformation of the utility function.
So it is better to normalize by the marginal utility to get a measure of inequality
aversion that is invariant to such transformations. This motivates the de…nition
given in equation (20). 3 With a function such a g that got constant relative
inequality aversion, inequality aversion is captured by one parameter a and the
larger a is, the more inequality aversion there is.
Now, let us get back to the distinction between consumption and utility
inequality aversion. This is best illustrated by considering three di¤erent cases.
1. Case 1: Utilitarian social welfare function and individual utility functions
with constant relative inequality aversion. A utilitarian social welfare
function is linear in the utility of each individual, so it exhibits no utility
inequality aversion: the utility of one individual is a perfect substitute for
that of another and the marginal social welfare of an increase in the utility
of some individual is the same as for any other individual (and typically
normalized to be equal to 1). If the individual utility function is of the
type
Uh = xh 1
1
1
(21)
then it exhibits consumption inequality aversion (and more so, the bigger
is 0). This consumption inequality aversion is inherited by social
welfare function, which in this case, then, exhibits consumption inequality
aversion but not utility inequality aversion. We can write the social welfare
function as
HX HX
SW F = Uh =
xh 1
1
1
: (22)
h=1
2. Case 2: Individual utility functions are linear in consumption (Uh = x h 1)
and the social welfare function is of the following type:
SW F =
h=1
HX (Uh) 1
h=1
1
: (23)
3 If you are familiar with the de…nition of risk aversion, you will see the similarity to that
concept.
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