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CHAPTER 23
Social
welfare
functions
The interest of the community then is

what? The sum of the interests of the
several membets who comDose
lt'
Jeremv Bentham
Whereas one can speak ofrlepositive theory
ofpublic choice, based upon economic
man assumptions, one must think of normative theories
of ptrblic choice, for there
are many views ofwhat the
goals ofthe state should be and how to achieve them. This
potential multiplicity has been the focus ofmuch criticism by positivists, who have
argued for a "valueftee" discipline. For the
bulk ofeconomics, it mightbe legitimate
.
to focus on explanation and prediction, and leave to politics the explication of the
goals of society. For the study of politics itself, in toto, to take this position is less
legitimate; thus the interest in how the basic"ralues ofsociety
are or can be expressed
through the political process. The challenge that normative theory
faces is to develop
theorems about the expression and realization ofvalues, based on generally accepted
postulates, in the same way that positive theory has developed explanatory and
predictive theorems from the postulates ofrational egoistic behavior. Part V reviews
some efforts to take up this challenge.
23,1
The BergsonSamuelson social welfare function
The traditional means for representing the values
ofthe community in economics is
touse a social welfare function
(SWF). The seminal paper on SWI's is by Bergson
(1938), with the most significant further explication by Samuelson (1947, ch. 8).
The SWF can be written as follows:
W
=
lT(2t,22,.
..,2,),
where
ll
is a real valued frrnction of all variabies, and the z;s and
I/
are chosen
to represent the ethical values ofthe society or ofthe individuals
in it (Samuelson,
1947,p.221). The objective is to define
aW
and set
ofz;s, ard the constraints
thereon, to yield meaningfrrl fust and secondorder conditions
for a maximum
IZ. Although in principle any variables that arc lelated to
a societyt wellbeing
(e.g., crime statistics, weather data, years of schooling) might be included in the
SWF. economists have focused on economic variables. Thus, the SWF literature
has adopted the same assumptions about consumers,
production functions, and so
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Social welfare functions
on, that underlie the bulk of economics and public choice and has made these the
focal point of its analysis.
The only value postulate upon which general agreement has
been possible has
been the Pareto postulate. This postulate
suffices to bring about a set of necessary
conditions for the maximization of
Il,
which limit social choices to points
along
the generalized Pareto frontier. The proof is analogous to thb demonstration that
movement fiom offthe contract curve to points on it can be Pareto improvements,
and the necessary conditions are also analogous. With respect to production, these
conditions are
\Xi/\hi
:... :
\Xi/\Vi
_
T,r
0Xy/01\1,
0X1/0v6

7,1'
where
3
Xi
/
0
V*i is the marginal product
of factor V^ tn the production of out
put Xi, and
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 Spring '09
 Roth

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