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Faculty Working Paper 93-0167
330
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1993:167 CX)PY 2
The Library
JAN
C
University ot
of the
1994
Illinois
of Urbana-Chain|>aign
Generalized Samuelson Conditions and Welfare
Theorems
John
for
P. Conley
Department of Economics
University of IlUnios
Nonsmooth Economies
Dimitrios Diamantaras
Department of Economics
Temple University
Bureau of Economic and Business Research
College of
Commerce and
Business Administration
University of Illinois at Urbana-Champaign
BEBR
FACULTY WORKING PAPER NO.
College of
Commerce and
93-0167
Business Administration
University of Illinois at Urbana-Champaign
October 1993
Generalized Samuelson Conditions and Welfare
Theorems
for
Nonsmooth Economies
John
P.
Conley
Dimitrios Diamantaras
Department of Economics
Generalized Samuelson Conditions and Welfare Theoremsf
for
Nonsmooth Economies
John
P.
Conley*
and
Dimitrios Diamantaras**
Revised: October 1993
t
Version 1.0
*
Department of Economics, University of
Illinois,
Champaign, IL 61820
** Department of Economics, Temple University, Philadelphia,
PA
19122
Abstract
We
give intuitive Samuelson conditions for a very general class of
economies. Smoothness, monotonicity, transitivity and completeness are
not required.
We
provide necessary and sufficient conditions for
efficient allocations,
all
including those on the boundary.
We
all
Pareto
also prove that
if
agents have a cheaper point, the supporting prices fully decentralize the
allocation. Finally,
we show first and second
welfare theorems as corollaries
to the characterization of efficient allocations.
1.
Introduction
Samuelson (1954, 1955) gave the
One
goods.
of his
main
results
first
modern study
of economies with
was calculus-based conditions
for
pubhc
Pareto efficiency.
These "Samuelson conditions" have since become one of the fundamental
tools for
understanding public goods economies. However, his work has several important
limitations.
which
In particular, he did not deal with the issue of corner allocations, in
at least
one type of good
is
not consumed at
all
by
at least
one agent. Given
that this
is
probably the typical rather than the exceptional case
omission
is
not
trivial.
in real
life,
his
Unless we can characterize corner allocations as well, we
must doubt the practical relevance of the studies based on Samuelson conditions.
Later, economists
efficiency conditions
as
assumed that the most obvious modification of Samuelson's
would be the correct ones
Campbell and Truchon (1988) point out
where some
efficient allocations violate the
for dealing
with corners. However,
an important paper, there are cases
in
Sajnuelson conditions, even as modified.
Campbell and Truchon conclude that the Samuelson conditions miss some
allocations,
and they provide a
different specification of the
which they claim are necessary and
private
good and a
finite
number
Samuelson conditions
sufficient for efficiency in
of public goods.
efficient
economies with one
They assume
differentiability of
the utility and cost functions, convexity of preferences and cost, and monotonicity
of preferences.
Unfortunately, the analysis of Campbell and Truchon
sumption that there
The
first
is
is
limited by their as-
only one private good and their need for differentiability.
assumption reduces the relevance of
their contribution to
an essentially
partial equilibrium domain. Requiring differentiability significantly reduces the class
their analysis
may be
applied. Further, their proof of suffi-
ciency contains an oversight (proof of
Lemma
1,
of economies for
which
page 247), which we explain
in the
conclusion. These observations motivate an approach to the problem using convex
analysis, in the standard fashion estabhshed
by Arrow (1951)
for
economies with
private goods only.
Such an analysis was offered by Foley (1970)
in the course of formalizing the
general notion of Lindahl equilibrium. However, he requires in his definition that
allocations be in the relative interior of the private goods subspace of the
consump-
tion set of each agent. Thus, corner allocations are not dealt with by Foley, either.
Khan and Vohra
(1987) generalize Foley (1970) to allow general preferences and
nonconvexities, but their aim
is
to present the second welfare
theorem and they do
not examine the refinements needed to deal with boundary allocations.
In this paper
of private
we provide efficiency conditions
and public goods, without assuming
for
economies with a
differentiability.
We
finite
number
so not require
that commodities be goods. This allows us to consider important real world cases
such as the one in which a public project (a garbage incinerator, for example)
benefits
some agents while imposing
are not locally satiated. In addition,
are complete or transitive.
Our
costs
on others.
We
require only that agents
we do not assume that the preference
analysis deals with corner
a unified way. Unlike Campbell and Truchon 1988),
(
and
interior allocations in
we do not need
to appeal to the
Karush-Kuhn-Tucker theorem, and our proofs are simple and geometric
We
relations
in nature.
develop the most general from of the Samuelson conditions in a simple and
operational form, and we further show the existence of fully (Lindahl) supporting
prices at
any Pareto
efficient allocation, for all
agents
who
are allowed a cheaper
point by the Samuelson prices corresponding to the allocation.
these efficiency conditions
we show
first
As
and second welfare theorems.
corollaries to
2.
The Model
We consider an economy with L
private goods
consumers, and
£,
^A and
F
firms.
We
use the convention
and
M public goods, / individual
J =
{1,
.
.
.
,
/},
and similarly
for
Superscripts are used to represent firms and consumers and subscripts
J-.
to represent goods.
Each agent
relation
will
is
not
X
is
(x,
endowment
characterized by an
y ) where x
We remark that
less
bound
E
C
over the consumption set
>-'
be written
goods.
i
is
=
R^"^
A
.
G R^, and a preference
u;'
typical
a bundle of private goods, and y
assuming the consumption
set to
consumption bundle
is
a bundle of public
be the nonnegative orthant
general than Campbell and Truchon's introduction of a nonnegative lower
for the
consumption of the private good by each agent, since we can always
translate the preferences in order to
make
to generalize the results in this paper to
this lower
bound
zero.
It is
also possible
bounded below, convex consumption
sets
at the cost of complicating the proofs.
We make
the following assumptions on
Al)
>-' is irreflexive.
A2)
>-' is
A3)
If (x,j/)
continuous (the
strict
(x,y), then for
)>-'
>-'
for all
i
G
T.
upper and lower preferred
all
A G (0,
1),
A(x,y)
+
sets are open).
-
(1
A)(x, y)
)-'
(x, y).
(Weak convexity)
A4) For
all
(x,y)
We
II
(x,y) G C' and
<
€
and (x,y)
all e
>-'
>
(x,y).
there exists (x,y) G
^
normalize supporting prices to
C
||
(x,y)
—
(Local nonsatiation)
sum
to one, but
do not assume that prices
are positive:
The
such that
three kinds of vector inequalities are represented by >, >, and
^.
I
m
e
)
I
Define the marginal rate of substitution correspondence for consumer
z,
MRS'
:
C ^^ n, by:
MRS'(x,y)
{{P.q) e
n
|
(x,y)
{p.q')
<
(p,g')
•
=
(i,y) V(i,y) G
C
(f,y)
s.t.
)^'
(x,y)}
.
Define also the weak marginal rate of substitution correspondence for consumer
WMRS'
:
i,
C ^^ n, by:
WMRS'(j,y) =
n
{(p,?) 6
I
(p,5')
•
(x,y)
<
(p,g')
•
{i,y)y{i,y) e
Note that the marginal rate of substitution
the
weak marginal
empty. The
rate of substitution set
WMRS
correspondence
proof. Also note that
then (x,y)
hand
if (p,
is
g)
if (p,
q)
is
s.t.
MRS'(a:,y)
(i,y) x' (x,y)}
is
.
always a subset of
WMRS'(x,y), and MRS'(i:,y) can be
never empty-valued, as we indicate in our
G MRS'( j,
y)
and the agent has income
a preference maximizing choice over the budget
set.
(p, q'
On
)
•
(x, y),
the other
G WMRS'(x,y) then we are only guaranteed that {x,y) minimizes
expenditure over the
set of
consumption bundles that are not
[Figure
In the
set
C
example depicted
in
Figure
inferior to (x,y).
1 here]
1,
the agent's indifference curves intersect
the public good axis with an vertical slope, and terminate at their intersection with
this axis.
A
Otherwise, the preferences are standard, satisfying
and furthermore
all
other assumptions
commonly made on
all
of the assumptions
preferences. At every
point on the public good axis, the weak marginal rate of substitution correspondence
has a singleton value of
preferred
set.
Since the
and the unique
line of
WMRS
is
set,
goods and
y
is
will
correspondence,
the marginal rate of
empty-valued.
represent each firm / E /" by a production set P-^
production plan
MRS
correspondence contains the
support intersects the preferred
substitution correspondence
We
In other words, the vertical axis supports the
(1,0).'^
be written {z,y), where
^ is
C R^
x
R^. A
a net output vector of private
a the output vector of pubhc goods.
Define the marginal rate of transformation correspondence for
P^ —*—^
typical
P-'
,
MRT-^
:
n, as follows:
MRT^(.^y) =
The comprehensive
hull of a set in
comp(Z)=
For
all
/ E
^ we
{{z,y) G
is
a nonempty, closed
B2) P^
is
a convex
B3) Pf
= comp(P^)
p
|
is
defined as follows:
3(i, y)
G
Z
s.t. (z,
y)
<(£,y)}.
set.
set.
(Free disposal).
define the global production set in the usual way:
P=
and we
x R^^
R^
assume:
Bl) P^
We
R^
R^ x
{{z,y)
eR^ x^^
'+
(z,y)
= J2i^^^y^) and
(.~^y^) €
P^ V/ e
J^ i
define the aggregate marginal rate of transformation correspondence
,
MRT
:
^^ n by
MRT(^,y) = |(p,g) 6 n
I
{p,Y^q')-{z,y) >
(p,
^
g')
•
(f, y)
V
(i,y) e
P
Properly speaking, we should have indicated all the elements of the supporting vector here, in
accordance with the definition of MRS and WMRS. However, in all discussions of examples we
only indicate the components relating to the goods consumed by the agent in question, to enhance
clarity.
We make
P
B4)
the additional assumption:
closed.
is
P
Notice that
and comprehensiveness from the individual P^
inherits convexity
sets.
An
C
allocation
X P^ X
X
•••
is list
P^
.
=
a
Let
((x\
.4
A = la
y^ ),..., (j^, y^), (c^, y^
The
set of feasible allocations:
e C^ X
X
x')
C^ X P^ X
^
and
•
•
x
X P^
=
y^
C^ x
y'
V
?
G
T
/
'
Pareto efficient allocations
set of
(r^, y^)) G
...
denote the
Y^z^ = X](w' /
)
is
defined as
PE =
{ae A\^ae
Let A^~^ denote the /
—
A^-^ =
We
i
1
^
G
RM J^ ^'
.9^
.6^) G A^~^ X
X A^~^
z's
share of the profits of firm /.
An
=
and
1,
^'
>
V
z
G
J
>
,
.
.
allocation
...
and price vector
librium relative to the
.
.
denote a profit share system for a private ownership economy by ^
,
.
Is.t. (F,y'') y^ {x\y^)}
dimensional simplex:
{9^
.
3je
As.t. Vi G J, (x',y') ^' (x',y') and
endowment
=
(a,p, g)
uj
where
G
A
x
G R^^^ and
^''-^
11 is
is
=
interpreted as consumer
said to be a
profit shares ^
G
Lmdahl
if
equi-
and only
if:
/ G
(p,E.9') € MRT^zf^yf).
a.
for all
b.
iorsil\^eIAp.q)eMRS\x\y')aIid{p,q'){x\y) = pu' + Zf^''^iP^T.^Q)
T",
Note that given the definitions of MRS' and MRT-^, and the
fact that local
nonsatiation implies that each agent will exhaust his income, these are equivalent
to profit
and preference maximization,
by the definition of an
spondence
LE
:
respectively.
Feasibility
is
already required
allocation. Define the Lindahl equilibrium allocation corre-
—>— A
R^^^ X
as follows:
^
LE{u,d) =
{a G
A
\
for
some
E H, (a,p,
(p, q)
q)
start with a simple statement of our
main
results.
conditions are necessary for an allocation a G
A
There exists a price vector
such that
a.
for
allfeJ"
h. for all
I
€l,
(p,
(p, g^
?')
Ef=i
{p,q)
(this
would be true
interior of his
.
.
.
,
for
g"
)
£
11
There exists a price vector
allfeJ"
b.
for
all
I
(p,
The
following two
to be Pareto efficient:
if
for all
if
i
E
X
there
is
a cheaper point than {x\y) in
every agent's consumption bundle was in the
the following conditions are necessary for a to be
set),
a Pareto efficient allocation:
for
8}.
^ MKT^{zf,yf),
example
consumption
a.
and
eWMRS\x\y).
Alternatively, for a E A,
C
,
uJ
Results
3.
We
a Lindahl equilibrium for
is
(p, g^
ELi ?')
,
.
.
.
,
g") E
IT
such that
€ MRT^(.-/, y^),
E I, {p,q) E MRS*(x',y).
Finally, in
a G
A
both cases the following two conditions are
be Pareto
to
efficient
There exists a price vector
(p, g^
.
,
.
.
,
g"
G
)
11
such that
for
allfeJ" (p^ELi?') ^ MRT^(c/,y/),
b.
for
all
£ J, {p,q) E MRS'{x\y).
I
if
we have the
We
an allocation
:
a.
Notice that
sufficient for
we assume
then MRT-* and
differentiability,
MRS'
are singletons and
familiar Samuelson conditions.
begin our demonstration of these claims by showing that private goods
must be nonnegative.
prices
Lemma
For
1.
MRT(2,y),
G
(z^y)
all
P
and
all
p such that there
exists q
with {p,q) G
p>0.
Proof/
Suppose
there
a private good
is
some
not; then for
{z,y) G
P
and p,q such that {p,q) G MRT(-,y),
G £ such that p( <
^
{zi....,ze-S....,ZL,y)eP. But
(p, X:. g')(~~i,
By
0.
•
•
•
,
free disposal, for all 6
~f-^
•
•
•
,~L, y)
>(p,
E.
>
?')(-
y)'
contradicting the definition of MRT(^,y).
The
following
lemma
states given
an allocation a G
{z,y) majcimizes profits over the global production set
only
{z\y') maximizes the profits of each firm / G
if
P
^ at
A and
prices {p,q) G
at prices (p,
E'
?') ^^
11,
^^^
these prices. This allows
us to state the subsequent theorems in terms of maximizing profits over the global
production
Lemma
i^iV) €
P
set instead of
2.
if
going to the extra step of considering each firm.
Given {z,y) G
and only
P
if for all
and
f E
{p,q) G
11,
{z,y)
{p,q)
^ there exists {z^
,
>
{p,q)
(2, y) for all
y^) such that {p,q)-{z^ ,y^)
{p,q)-{zf,yf)forall{zf,yf)ePf andZfi^^,y^) =
i^^y)-
>
Proof/
1.
Suppose not, then
Necessity:
(~^y^) >
[p.q)
but for some
(5^-y^) for
ip^q)
G P,
(5, y)
(p, 5)
(c, y)
•
a collection of production vectors
However, by hypothesis
for all
= E/(P' ?)
implies (p,g)-(2,y)
<
J^,
there exists {z^ ,y^) such that
(r/,y/) G
all
(p, g)
(5-^,y-^)
/ G
^
/ ^
for all
6
•
P^ and E/(~^/) =
(c, y).
P-'^
^
definition, there exists
such that X]/^^'^-
{p,q)-{z^ ,y^
(-^' y^)
By
)
>
(p,
(p>9)-(^'y)
(-'?/).
q)-{z^ ,y^
=
~
^'^)
Ylfip^q)
But
).
'
i-'U)this
i^^ ^y^
)^
a
contradiction.
2.
Sufficiency:
(p, 5)
^
Suppose
-(r.y) for
J^:-'^, y-'^)
(p, g)-(2/'
iP^q)-
=
y/'
,
<
P
such that
(p, ^)
but for some
y^'
q)-{zf'
(p,
,
T^f^fi^^ ^y^)
+
{P^q)
).
/'
G ^, there exists (z^
(c,y)
>
that
,y^)E P^ such
that
But then E/^/'(-~^. y^) + (^-^'
{^^\y^') >
•
P^ such
{z,y) G P, and plan for each firm {z^,y^) G
all
{z, y),
)
not, then there exists {z,y) G
y^'
,
)
G P, and
iP^q) -Jlfi^^^y^)^ ^ contradiction.
•
We now
Theorem
give the
1.
If a
G
A
vector {p,q\...,q'') G
for all
I
necessity theorem.
first
is
H
a Pareto efficient allocation, then there exists a price
such that (a) (p,^!.?') ^ MRT(X;,(-?^' -^''),y) and, (h)
G J, (p,9) G WMRS'(x',y).
Proof/
Following Foley, we define an
artificial
production
set in
which public goods
are treated as strictly jointly produced private goods:
AP =
AP
is
{{z,
y\...,y')\y'=...y' = yaxid{z,y)eP}.
closed, convex,
properties. Next
we
and nonempty
as a consequence of
P
possessing these
define the socially preferred set of the allocation a:
I
SP{a)=
rJ\ r- TO L ^. IQ) [
{{z,y\...,y^)eR^xR
= E(i'--')1=1
V
The
6 J, V (x',y') G C',(x.,y)
^
I
{i\y') and 3; G
>^'
(i^y^)
s.t.
>^^
(x,,y)
^.
and by continuity and nonsatiation
socially preferred set inherits convexity,
it
has nonempty interior.
a.
Since a
Pareto
is
by assumption, SP{a)
efficient
Minkowski Separation Theorem (Takayama (1985,
vector {p,q^
.
.
,q^
.
)
For
all
(ii)
For
all (s,
with
||
y\
.
.
.
,
p
||
<
and a
oo,
+ ^^q'
z
-y
y^) G closure( 5P(a)), p
.
.
.
Therefore, for
,
y^) G
all
AP.
y
the
there exists a price
such that:
f =
>
=
y^
=
y.)
r.
6 closure(5P(a)). By hypoth-
and
(i)
•
Then by
0.
that p- z
(ii)
+
X],^'
'
H
= ^
(i,y) G P:
P-
it is
from
follows
It
)
r,
(where
+ E, g'
i
•
p. 44)),
scalar
<r
continuity and nonsatiation, (c,y\...,y
esis, (2, y^,
Since
^
(5,y\...,y^) € AP, p-
(i)
By
,
AP =
fl
+
z
^q'
-y
=^r>p-z + ^q'
-y.
possible to renormalize these prices to be elements of H, this establishes
part (a) of the theorem.
b.
Now
suppose that part (b)
such that [x^ ,y^) y^
(a:-',
^(x' -
0.')
false.
is
y)
+
and
(x^
Then
(p, 9-')
-
u;^
),
•
[x^ ,y^)
yS
.
.
.
,
I
6
for all j
<
y^
.
there exists
{p,q^)
.
.
,
y^
)
•
+ p-ixJ
-LjJ)
+ Yl 9'
G SP{a)
-y' ^<i^ -y^
<(p,9\---,/)-fe(^'-^')'J/''---'^J'
a contradiction to
(ii)
above.
10
j-'
,
y-'
)
[x^ ,y). Hence,
and
p Y^{x' -Lo')
(
G
C-'
As a corollary
particular,
prices for
firm
is
we show
some
profit
to this
that
set of
we
state a version of the second welfare theorem.
we can
In
decentralize any Pareto efficient allocation through
endowments and
maximizing under the
such that the production of each
profit shares
prices,
and each agent's consumption bundle
minimizes expenditure over the weakly preferred
set.
This
is
not quite the
same
thing as decentralizing the allocation as a Lindahl equilibrium since agents are not
necessarily maximizing preferences over the budget set.
slightly stronger conditions as needed.
We
show
To
get this stronger result,
this below. See
Debreu (1959)
for
details.
Corollary 1.1 (weak second welfare theorem)
cation, then there exists a price vector (p, g^
and a proEt share system 6 such
that (a) (p,
,
.
If a
.
^^
,
.
E
g") €
q*)
A
11
is
a Pareto efficient allo-
an endowment vector
€ MRT(^j(x' —
u,?'),
lj,
y), (b) for
allieI,andall{i\y)y^{x\y),{p.q')-ii\y)<p-^'+Zf^''^iP^E,<l)-i^^^y^)
ai^d, (c)
^^iOt
=
^,'^t
Proof /
We know
by Theorem
1,
there exist prices {p,q) G
MRT(X;,(j:' -u;'),y) and (b) for
all
i
H
such that (a)(p,
£ I, {p,q') € WMRS'(x',
^(p,g').(x',y')
j/).
profit shares in a
It
only remains to show that we can redistribute
way
g')
G
Notice that:
=
In words, the total cost of consumption equals the value of the
the profit shares.
^^
endowment
plus
endowment and
that satisfies (c) such that the cost of the Pareto efficient
11
consumption bundle (x',y)
But
prices.
clearly this
when we vary
over the
full
is
each agent equals the implied income under these
for
possible since total income to society does not change
the distribution,
range of
we can continuously vary the income distribution
possibilities,
enough endowment so that when
and we know from the above that there
it is
fully distributed, society's
We
Next we give a second necessity theorem.
require that
all
strengthen the hypothesis to
us to conclude that there will exist supporting prices in the
WMRS.
exactly
budget balances.
agents have a cheaper point in the consumption
of each agent, instead of just the
is
This allows
set.
MRS
correspondence
This means that the prices are fully
decentralizing.
Theorem
(a)
p
2. If a
>
x'
0,
E
or (b)
A
a Pareto efficient allocation, then for every
is
3m
<
s.t. ql^
0,
or (c) 3
Theorem
are the prices established by
m
s.t. q]^
we have
1,
i
and ym >
>
E
X
sucii that
where
0,
p, q'
that {p,q) E MRS'(j',y).
Proof /
(a)
Suppose that
for
some
i
E T, p-
x^
>
and
implies that there exists (x', y') E C' such that (x',y')
(p, g')
•
{x\
y').
Since p
>
by
implies that there exists £ E
Denote the open
line
Lemma
C such
1,
and
that pe
>
>
x'
I
By
{x\y) and
>-'
segment between two points as
3X E (0,1) and (x\y')
=
C\
p- x'
(x*,y').
€
i:((x',y),(x',y')),
we have
follows:
=
\{x\y)
(i',y') X' (x',y)
+
(1
-
A)(x',y')}
and
(p,g')
For (x',y') close enough to (x',y) (A close enough to
12
>
0.
.
the convexity of preferences and the linearity of the budget constraint, for
(i',y')
>
(p, g')-(x', y)
because {x\ y) E
and x\ >
L{{x\y),{x\y'))
{(f',y')
^ MRS'(j:',y). The latter
(p, g)
•
1),
(x',y)
x^
>
>
0.
all
{p,q')
By
•
the
continuity of preferences, there exists
<
{i\y^)
(p. ?')
(J^*'
•
(b)
such that (x\
y)
>
for
.
.
same contradiction
(c) Finally,
(PiQ^)
{x\
[p, q^)
such that {x\y\,
.
some
£ J,
i
suppose that
of (a) above, with
3m
By
s.t. ql^
C
£
as in Figure
1,
.
.
(ii)
in the
<
and
.
,
.
.
i'^, y'
,
<
x\,y^)
)
>-
(p, g')
proof of Theorem
•
1
{p,q^) ^
MRS'(x',y).
such that (j',y') x' (x\y) and
)^'
(j',y).
Since g^
y^
(p, g')
for
latter
•
some
i
3m
€ X,
s.t.
imphes that there
(x',y)
>
(p, g')
in the place of x\
•
(x', y').
and 5^
<
0, this
>
Q
e
>
leads to the
that the extra assumption
is
good
of the private
and agent
for
q\^
exists (x',y')
We
C
E
and
such that
can now mimic the proof
required to obtain the
full
support
an economy with two agents, one
linear technology,
each agent. Agent
2 has translated
and ym >
in the place of pi.
and one public good, one firm with one-to-one
ment of one unit
e,
.
the continuity of preferences, there exists
illustrated in the following example. Consider
private
—
,x\
.
e,
as before.
{x\y) and
The reason
y').
+e,...,y)^)
,y\^
i MRS'(x',y). The
(x', y') >-'
.
—
x\
as in that proof.
Suppose now that
{x\
•
{x\,.
•
leading to a contradiction to
y),
latter implies that there exists (x',y')
(p, g')
is
>
there follows {p,q^)
0,
same manner
in the
The
>
Since pe
(x', y).
e
and endow-
has preferences exactly
1
Cobb-Douglas preferences such that the
slope of agent 2's indifference curve at (x^, y^
)
=
(1/2,3/2)
is
—1.
Then
the alloca-
WMRS^O,
|)
contains only the vector (1,0), which intersects the strictly preferred set of agent
1.
tion
(x^y^x^y^2^y^) =
(0,
|,|,
|,-|,|)
Therefore, the Samuelson prices arising from
and
2.
this failure occurs for agent
1
who
is
Pareto
Theorem
efficient,
1
but
are not separating prices,
violates all three of the conditions of
Theorem
This allows us to state a stronger second welfare theorem.
Corollary 2.1
("strong
cation such that for
all
second welfare theorem)
agents
i
E
2",
If a
G
A
is
(x', y) is in the interior
13
a Pareto efficient allo-
of C\ then there exists
a price vector {p,q^
,
.
.
,q"^)
.
E
U.
an endowment vector u, and a profit share system
9 such that a € LE{Cj,d)'and Xlj^^i
—
Z]i*^'-
is
in the interior of
Proof/
Since for
rem
2
all
agents
G X,
z
,
y)
satisfied. Therefore, there exist prices (p, g)
is
MRT(^f^i(x' -^'),y) and
all
(x'
(b) for all
i
when he maximizes
his preferences while
we know from the argument given
endowments and
divide
social
endowment
Now we
Theorem
is
in the
11
the hypothesis of Theo-
such that (a)(p, ^,_i
G I, (p,g) € MRS'(x^y). But
firms profit maximize under the prices, and (b)
(x',y)
€
C,
having income {p,q')
proof of Corollary
1.1 that
G
)
means
(a)
means each consumer
«?'
chooses
i
{x\y). But
it is
possible to
and the
profits so that each agent has exactly his income,
exactly exhausted.
give our sufficiency theorem.
3. If a
£
A
is
an allocation and there exists a price vector {p,q^
such that (a) ip^YlUi^') ^ MRT(X;[=i(a:' -u^'),y) and (b) for
MRS*(x',y), then a
is
Pareto
all
,
.
.
,q
.
)
G I, (p,g) G
i
efficient.
Proof/
Suppose that the hypotheses of the Theorem are met but a
efficient.
for
no
z
Then
G
(x-^, y-') >-^
J
is
there exists a (Pareto dominant) feasible allocation a E
it
(x',y').
the case that (x',y')
Then by
(b)
>-'
(x',y')
and
and summing up over
for
all
till
some
(a),
11
t
14
i
j
not Pareto
A
such that
G X, we have
agents,
^p-x' + ^g'-y >^p-x'+^g'-y.
But by
is
•
(i)
Since
(ii)
contradicts
Finally,
we
Corollary 3.1
(i),
the proof
is
get the first welfare
If a
G LE{u},d), a
finished.
theorem as an immediate consequence of
Pareto
is
this.
efficient.
Proof/
By
the definition of Lindahl equilibrium, there exists a price vector {p,q^
such that (a) (p,Ef=i9') ^ ^'IH-T(Ef=i('^' -^').y) and (b) for
MRS'(x',y). But then by Theorem
4.
The
of
3,
Pareto
Some comments on
1.
The
first
Condition
GOC in
ferentiability assumption)
as
shown
.
.
,q^
G I, {p,q) G
efficient.
the literature
1988) occurs on page 247, in the proof
may be
negative.
Campbell and Truchon (1988) attempts
pushed against the public good
In such a case,
(
i
.
inequality of the last sequence of inequalities in that proof
does not hold, because some the dx,
WMRS,
is
oversight in Campbell and Truchon
Lemma
unity.
a
all
,
if
MRS
axis
by weighting their (unique, under their
by a nonnegative
we allow
to deal with agents
coefficient that
the coefficient to run from
to
can be
1,
we
less
dif-
than
trace our
in Figure 2.
[Figure 2 here]
Khan and Vohra (1987) have general assumptions on preferences and
for nonconvexities in preferences
and production, but they require
15
all
they allow
public goods
to
be desired by
introduction
all
falls
agents, so that the example with the incinerator mentioned in our
They prove a version
outside their coverage.
theorem employing a notion of supporting vector
to our
WMRS(x',y) (Khan and Vohra
set equivalent,
under convexity,
1987, page 236).
two notes on papers that are tangentially
Finally,
of the second welfare
relevant.
Saijo (1990) ad-
dresses a quite different point arising from Campbell and Truchon than
namely, he shows that the robustness of boundary Pareto
served by Cambpell and Truchon
economies, since
ter 3) contains
it
also
happens
not a
is
phenomenon
an extension of Foley's (1970) results
to
do;
efficient allocations ob-
specific to public
exchange economies.
in
we
Manning
good
(1993, chap-
economies with local public
goods, using assumptions based on Foley's, such as constant returns to scale and
ruling out the private goods boundary.
5.
Arrow, K.
"An extension
(1951):
nomics," in
J.
Neyman
sium on Mathematical
Berkeley
:
(ed.),
References
of the basic theorems of classical welfare eco-
by Proceedings of the Second Berkeley Sympo-
Statistics
and Probability.
University of California Press, pp. 507-532/
Campbell, D.E., and M. Truchon
(1988):
"Boundary optima and the theory
of public goods supply," Journal of Public Economics, 35, 241-249.
Debreu, G.
(1959):
The theory of
D.K.
(1970):
"Lindahl's solution and the core of an
Foley,
value. Yale University Press.
economy with pubhc
goods," Econometrica, 38(1), 66-72.
Khan, M.A. and R. Vohra
rem
(1987):
"An extension
to economies with nonconvexities
16
of the second welfare theo-
and public goods," Quarterly Journal of
Economics, 223-241.
Manning, J.R.A.
(1993): Local public goods: a theory of the first best. University
of Rochester Doctoral Dissertation.
Saijo, T. (1990):
"Boundary optima and the theory of pubHc goods supply: a
comment," Journal
Samuelson,
P. (1954): "The pure theory of public expenditure," Review of Eco-
nomics and
Takayama, A.
2nd
of Public Economics, 42, 213-217.
Statistics, 36,
(1985):
387-389.
Mathematical Economics.
edition.
17
Cambridge University
Press,
Figure
1
Campbell+Truchon's
*:
The
MRS
horizontal edge of
WMRS
Figure 2
is
not in
MRS
HECKMAN
IX
BINDERY
|S
INC.
JUN95
,una.To.P.eas^ N;
MANCHESTER,