max
(X
hg
; V
hf
; X
g
; r
gf
)
U
1
X
lg
;V
lf
s:t:
U
h
U
h
X
hg
;V
hf
h 2, ...,H
X
g
f
g
r
gf
g 1, ...,G
P
H
h1
X
hg
X
g
g 1, ...,G
P
H
h1
V
hf
P
G
g1
r
gf
f 1, ...,F
The pareto-optimal conditions follow directly from the Wrst-order condi-
tions of this constrained optimization problem. We will derive them later on
in the chapter.
EQUITY: THE SOCIAL WELFARE FUNCTION AND OPTIMAL
DISTRIBUTION OF INCOME
Although the model as it stands is suYciently detailed to analyze the neces-
sary conditions for allocational eYciency, it is entirely neutral with respect to
any equity norms. Chapter 1 described two types of equity, process equity
and end-results equity. The model is silent regarding process equity. This is
not so troubling in a social planning context, however, because the planner
simply dictates all economic decisions. Process equity norms such as equal
opportunity and social mobility are far more relevant in a market context, in
which the degree of process equity depends primarily on the structure of the
individual markets. Equal opportunity and a reasonable amount of social
mobility are likely to be achieved if markets are highly competitive. Market
power and other kinds of market imperfections are the chief enemies of these
norms.
The same cannot be said about end-results equity, the quest for a just
distribution of income. We saw in Chapter 1 that end-results equity is a
fundamental issue for any society, even when all the technical and market
assumptions for a well-functioning economy hold.
The baseline, social planning eYciency model described above illustrates
the end-results equity problem in the following manner. The Wrst-order con-
ditions for the constrained optimum of the model solve for a single allocation
of resources, a single point on the utility possibilities frontier. But the con-
straints imposed upon utility levels of persons h 2,..., H, the
U
h
, are
entirely arbitrary. Placing at least one of these consumers at a diVerent utility
level and solving the model again generates a diVerent allocation of resources,
so long as the new constraints permit a feasible solution (U
1
() 0). Since the
utility constraints can be reset in inWnitely many ways, solutions to the
constrained optimum problem generate an inWnity of feasible solutions in
2. A GENERAL EQUILIBRIUM MODEL FOR PUBLIC SECTOR ANALYSIS 39
general, all points on the utility-possibilities frontier. Furthermore, the model
as it stands has no way of choosing a best allocation among these allocations.
According to the pareto criterion, all allocations on the frontier are optimal
and therefore equivalent. Pareto optimality is an extremely weak normative
criterion in this sense.
The inability of the pareto criterion to choose a best allocation is a
glaring weakness for a normative theory of the public sector. For instance,
the following allocations are equivalent in a two-person economy in terms of
the pareto criterion: Person 2 receives almost all the goods and services, and
person 1 almost nothing; each person receives an equal allocation of the
goods and services; person 1 receives almost all the goods and services, and
person 2 almost nothing. The baseline model is completely neutral regarding
these outcomes.
Societies are typically not so neutral, however. They embrace a set of
end-results equity norms and devise some method of ranking the possible
outcomes according to these norms. At the very least, most societies express a
concern about the extremes of wealth and poverty.
The Bergson±Samuelson Social Welfare Function
Because most public sector economists believe economic analysis is properly
concerned with end-results equity, they have seen Wt to include a representa-
tion of distributional rankings in their models. The model requires a function
that indicates the desirability from society's perspective, the social welfare, of
all the possible distributions of individual utility or well being. The function
almost universally chosen for this purpose is the so-called Bergson±Samuel-
son individualistic social welfare function,
4
Wrst described by Abram Bergson
and Paul Samuelson in the late 1930s:
W WU
1
X
lg
;V
lf
, ...,U
H
X
Hg
;V
Hf
or simply
W WU
h
X
hg
;V
hf
(2:5)
with qW/qU
h
> 0, for all h.
The social welfare function is said to be individualistic because its only
arguments are the individuals' utility functions. That is, W( ) measures the
social welfare attained in each possible state of the economy by considering
4
After Abram Bergson and Paul Samuelson, who Wrst described the function. Samuelson
used this construct in his 1954 article, ``The Pure Theory of Public Expenditure,'' referred to in
footnote 1. Refer to Samuelson's lucid discussion of the social welfare function in P. A.
Samuelson, Foundations of Economic Analysis, Atheneum Publishers, New York, 1965, pp.
219±230. See also A. Bergson, ``A Reformulation of Certain Aspects of Welfare Economics,''
Quarterly Journal of Economics, 1938.
40 EQUITY: THE SOCIAL WELFARE FUNCTION AND OPTIMAL DISTRIBUTION OF INCOME
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