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Him
.M415
no© 3
working paper
department
of economics
Lindahl's Solution and the Core of
An Econouiy with Public Goods
by
Duncan K. Foley
Number
3 -
October 1967
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
Lindahl's Solution and the Core of
An Economy with Public Goods
by
uuncan K. Foley
Number
3
-
October 1967
C^Cf^-f
I
This paper was supported by the National Science
Foundation (grant number GS 1585 )
The views expressed
in this paper are the author's sole responsibility, and
do not reflect those of the National Science Foundation,
the Department of Economics, nor of the Massachusetts
Institute of Technology.
•
rvo »"5
MAY G
m.
i.
r.
1988
LIBRARIES
.
LINDAHL'S SOLUTION AND THE CORE OF
AN ECONOMY WITH PUBLIC GOODS
Introduction
I.
Since Samuelson's clear statement of the theory of consumption
externalities (public goods)
[7],
there has been substantial progress
in sharpening analytical tools for tax and expenditure problems.
An
important part of this progress has been the rehabilitation and
reconstruction of Lindahl's
problem
[6]
quasi-demand solution to the taxation
by Johansen [4] and Samuelson [8].
More recently the theory of the core of an n-person game has been
applied elegantly to the problem of economic general equilibrium
[9]
The purpose of this paper is to state rigorously the nodern theory of
resource allocation with public goods, and to study the relationship
between Lindahl's solution and the core.
Section II introduces assumptions and definitions, including the
notion of "public competitive equilibrium
71
of
[3].
Section III
establishes the relation between marginal rates of transformation and
marginal rates of substitution at Pareto optimal allocations of the
economy.
This is a reconstruction of Samuelson's first paper [7].
Section IV defines and establishes the existence of Lindahl equilibrium.
Section V defines the core.
Section VI consists of a proof that a
Lindahl equilibrium is in the core.
)
-2-
Definitions and Notation
II.
The economy has m public goods and k private goods.
of public and private goods is written (x-,...,x
;
in
jl
A vector
y-,,...,y.) = (x;y).
J&
There are n consumers, distinguished by superscripts.
A.
Consumption
Each consumer chooses a point in a consumption set X
,
on which
there is defined a complete and transitive preference ordering ^.
owns an initial endowment of private goods w
,
and
.
n
i
A. 1 X = EX.
<
has a lower bound for —
i=l
A. 2
X
and has an interior in the
is closed and convex for each i,
private good subspace.
A. 3 If x
e
X
there is x
A. 4 For every x
{x
X
e
x.».
^
I
l
'
x
A. 5 If
A. 6
^. x
the sets {x
x\
l
l
(x;y)
>_
+ bx
then ax
(x ;y)
e
)
^
X
x.^. x.}
|
<
(x";y)
and
(Continuity.)
.
with y
in X
(Non-satiation.
.
where a,b
x
.
then (x;y) ^.
,
^. x
are closed in X
}
There is a point (x;y
A. 7 If
B.
X
e
with x
X
e
w
and a + b =
>
1.
(Convexity.)
.
for all
i.
Production
Production is denoted by a vector (x;z) with inputs negative and
The set of all technically possible production plans
outputs positive.
is called Y.
That is, if (x;z) and (x;y)
B.l Y is a closed, convex cone.
a,b
B.2
e
>
0,
for all
e
Y.
at least one z.
<
0.
Y and
Y.
B.3 If (x;z)^
!*•
As is
then (ax + bx; az + bz)
e
Y,
e
customary, x
j
;
x
12
>_
x
12
>
x
1
means x.
means x
12
>_
x
2
> x.
_
..
for all i;
but not x
12
= x
.
x
2
I
2
1 >
_ x means x. > x.
i
3-
B.4 There is (x;z)
Y with x.
e
>
0,
= l,...,m.
j
(Possibility of
producing public goods.)
B.5 If (x;z)
Y and
e
= x. when x.
x".
J
J
then (x;z)
C.
when x.
J
<
0,
3
(No public good is necessary as a production input.)
Y.
e
but x. =
>
-
J
Allocations
C.l An allocation is a vector of public goods x and a set of n vectors
of private goods (y ,...,y
/
(x;y
i\
,,i
—
with y i
•
X
e
)
*.i_
<
y
such that for all
)
there is
i
i
.
C.2 A feasible allocation is an allocation (x;y ,...,y
n
i
i
X
[x;.E (y -w )]
e Y.
1
C.3 A Pareto optimum is a feasible allocation (x;y
there is no other feasible allocation
K
1
(x'jy
)
such that
)
(x^ 1 )
for all
(x";"y
,
.
.
,
.
.
,y
.
.
,y
such that
)
with
)
i.
C.4 A public competitive equil ibrium is a feasible allocation
(x;y
,
(t
1
, .
.
.
,...,t
n.
.
i_
^(y
p*[x;
i
\
,
with
)
n
a)
price system
a
),
,y
i
= P *w
p
-w
i
x
*x =
Si
E,t
1=1
.
p =
(p
and a vector of taxes
),
;p
.
such that:
_
i
^_p-(x;z) for all (x;z)
)]
- t
i
—
-r /
(x;y i\)
and if
i
^.
r
±
i\
e
/
(x;y
),
Y;
—
then p -y i
-1
b)
P
c)
there is no vector of public goods and taxes (x;
t
n —
—
p «x = .I,t
1=1
x
_i
y
(x;y
y
)
^.
r
i
such that for every i there exists y
(x;y
)
and p »y
y
—<
>w
p
- t
i
>
p
,...,t
)
-y
;
with
with
.
y
Public competitive equilibrium involves profit maximization by
producers, preference maximization under the after-tax budget constraint
by consumers, and the impossibility of finding a new public sector with
taxes to pay for it that appears to every individual to leave him better
off.
See Foley [3] for a discussion of this definition and a proof that
.
-4-
a public competitive equilibrium is a Pareto optimum.
III. Prices
The prices of public goods in a public competitive equilibrium
Samuelson
are marginal rates of transformation.
has shown that
[7]
at a Pareto optimum the marginal rates of transformation for public
goods are equal to the sums of marginal rates of substitution.
is the essence of the following
This
theorem.
There is a separate public good price vector for each individual.
Every individual facing his separate price vector "demands" the same
The sum of the individual
vector of public goods as every other.
price vectors is a social price for the public goods which is the
correct guide for profit-maximizing producers to follow.
Notice the
symmetry: in the case of private goods everyone faces the same prices
and each person chooses a different bundle of goods, while in the case
of public goods everyone chooses the same bundle but faces
different prices.
A.
a
Theorem
If
:
(x
price vector p =
a)
y
;
(p
for all (x,z)
Proof
(x;y
If
b)
...,Px
.
1
y±
(xr.y
),
[
(x
,
.
.
.
,x
;
z)
|
x
F is a convex cone since if
and a,b
>
such that
>
)
w 1 )] > (j^p* 1
1 -
Py )'(x;i)
;
Y.
e
X
)
p
;
.^(y
there exists
1
p^-x +
then
P 1-X
>
Py-y
x
+
P
-
y
y
•
Let
:
F =
,
.
is a Pareto optimum,
)
.
Py )-[x;
(j^p/;
,y
,
0,
then (ax
=...= x
—1
(x
= x and
— rx
,
.
.
.
+ bx ,...,ax
,x
;
—
z)
+ bx
(xrz)
a
and (x
;
Y]
e
1
,
.
.
.
~n
,x
az + bz) has
"
;
z)
e
F,
-5-
-1
ax
,
-1
-n
-1
-n
~n
-n
=...= ax + bx because x =...= x and x =...= x
*1
,
+ bx
,
(ax + bx;
az + bz)
Y because
e
(x;z) and
(x;z)
,
and
Y and Y is a convex cone.
e
Let
D = {(x
x
z)
;
D is convex since if (x
with a + b =
(ax
+ bx
then ax
1,
.
,
.
.
.
z)
;
.
,
r,
)
with
I z
,x
+ bx
+ w
+ bz
az
;
z =
|
.
.
and(x
)
,
.
+ bx
,ax
(x;y
(x
.
,
z
.
,x
+ w
r
z
D and a,b >
1
z)
;
)
.
(x;y
)
}.
az + bz) will have
;
by convexity of preference,
and so will be in D.
D and F have no points in common, since if they did (x; y ,...,y
By the Minkowski separating hyperplane
would not be a Pareto optimum.
theorem (for one proof see Karlin [5]), there exists
(p
X
,...,p
p
;
X
price vector
and a scalar r such that
^
)
a
J
n
(1)
-1
-n
for all (x ,...,x
(2)
-1
-n
for all (x ,...,x ; z) e D,
;
z)
e
i
-
Z,p x *x + p -z
F,
n
i
-5-iP
By continuity of preference (x,...,x;z)
e
D,
-i
,x
+
r;
<_
and
p
•
z
>_
r.
the closure of D, and is
in F so that
n
.
E
i
..
p
•
i=l*x
x + p
for all (x;z)
Since
y
z =
r>—
n
_
i _
*x + p -z
.£,p~
1=1' x
*y
Y, which establishes part a)
e
e
•
F,
r
>_
of the theorem.
and if r were positive the activity
(x,...,x;z) could be expanded indefinitely giving a higher profit
and contradicting (1) so that r =
0.
Points with very
Suppose p had some negative component.
large amounts of the corresponding commodity would be in D by
monotonicity, but would have smaller value than (x,...,x;z) which
contradicts (2).
Therefore p
>_
=0.
Suppose, however, that p
i
0.
y
Some p
x
>_
0.
Since production
)
-6-
of all public goods is possible with no public good input, there
would be a point in F with positive profit, which contradicts (2).
Therefore p
->
J
0.
>
-h.
y )
,-h
_
Suppose (x
;
'
h
.
(x
.
y
)
k
-k
k
-k
where y = y and x = x for k
.Z.p
i
i=l*x
-x
x
+
p
*y
-[.^(y
i=l w
1
L
,-1
h.
;
-
w
The point [x
.
4-
-n
, .
.
.
,x
Z
,-i
(y
-
w )].
_
.
;
w
i..
)]
h is in D so
1
>
—
)]
.S.p
x
i=lrX
>x
+
P
K
.
y
[.E, (y
1=1 '
1
1
Since all terms are the same on both sides except those corresponding
to h, it must be true that
px
h -h
x
-h
,
*
h
p y' y
-
p
'
>_
0,
there is another point in X
Along the line between this point and (x ;y
value than (x;y
but near (x ;y
),
preferred to (x;y
another point with all private
Since there is in X
Suppose equality held.
goods smaller and p
k
P y' y
'
x
)
)
x
h -h
'X
+ pr
This corresponds to a point in D with smaller
).
-h
,
y
-y
J
all points have lower
by continuity there will be a point
value than (x,...,x;z) which contradicts (2).
p
r
with lower value.
h
>
p
*x
-x
+ p »y
Therefore
h
y
and part b) of the theorem is established.
n
I
t
= p
leave it to the reader to verify
that p = .E,p
:
i=l^x
*x
•
(w
- y
i
and
also satisfy the definition of public competitive
)
equilibrium in II.
C. 4.
IV.
Lindahl Equilibrium
Corresponding to each Pareto optimum there is a total tax on
each person
t
= p
•
(w
- y
)
which may be positive or negative.
There is also a value for total
-7-
consumption
i
p
r
x
*x
+ p «y
i
y
which may be greater or smaller than endowment income.
[8]
Samuelson
proposes to call the difference between this consumption value and
endowment income a lump-sum transfer
_i
L
= p
r
x
i
+
'x
p
*y
i
- p
y
*w
:
i
.
y
As Johnson [4] explicated clearly, it was Lindahl's idea [6]
a kind of mock competitive equilibrium in the public sector.
to find
On the
assumption that the endowments of the society have already been
redistributed to achieve an ethical optimum,
this competitive equilibrium
will also be the social welfare optimum.
Competitive equilibrium requires that all consumers be maximizing
satisfaction at the income they get from their own endowment.
In other
In the language of the
words all the lump-sum transfers must be zero.
"benefit theory," the value of public goods received by each individual
is equal to the total tax he pays.
A.
D efinition
n
1
A Lindahl equilibrium is a feasible allocation
and a price system (p
a)
-
\
o)
n
i
-?iP
v
X X
( n
l--
c
if
•
r-i-
(x
:
;
P, ,)*
,
[
n
x ;,-?i (y
X~~
y
~i\ v
>.
) r
y
j
1
B.
...,p
/
(x:
;
)
i\
)
_>
i
^i
then pr
x
,
.
.
.
,y
)
(
~i
*x
Lp X i
X~~X
+
i
p
;
~i
p
*y
y
--
--
n
- w )]
y
such that
>_
ii
X
y
>
p
(x;
)-(x;z) for all (x;z)
^
>
p
i
x
-x
+
i
p
*y
y
i
= p *w
y
Existence
To prove the existence of a Lindahl equilibrium the following lemma,
adapted trivially from Debreu
Lemma:
[2]
Y;
e
v
is necessary.
Given the assumptions II. A. 1-5 and II. B. 1-3, there exists in a
i
.
-8-
private good economy a feasible allocation (y ,...,y
such that
vector p f
[-^(y
1
w
"
> p-z
a)
P'
b)
for every i, if y
Theorem
)]
^
for all
z
Y;
e
then p -y
y
Under assumptions II. A and
:
and a price
)
>_
= p
-y
p
-w
.
there exists a Lindahl
II.fi
equilibrium.
Proof
The strategy is to construct a private good economy to which the
:
lemma applies and show that the quasi-equilibrium of this economy is a
Lindahl equilibrium.
Extend the commodity space by considering each consumer's bundle
In this nm + k
of public goods as a separate group of commodities.
space, the set F meets the requirements of II. B. 1-3.
by writing zeroes for all public good
Extend the sets X
components not corresponding to the
consumer.
i
These sets meet the
requirements of II. A. 1-5.
From the Lemma, there will be a vector (x ,...,x
prices
1
1
/
(p
'x
,...,p
n
.
p
:
x
'
)
,
_
(
if 1
P
(x;z)
(2)
(x
i
P.,
a
y
in Y.
-ix v
-i
if
-i
**
y
;
+
)
p
y
)
and
,
v
^.
-y
.^(y
1
w
-
=...= x
(x
-i
,
,...,y
y
P )'tx;
;
x
y
sucu tuat
f
1
(1)
;
,
(x;y
)
i
_
> p
^
x
a
-x
1
)]
>
(j^p/;
Py )'(x:i)
for all
= x as a consequence of the definition of F.)
.
len
the
+
>y
p
i
= p
y
-w
i
.
y
An argument similar to the one based on monotonicity used in
section II will assure that
p
*"
->
and
n
K
y
>
-
0.
-9-
Part (2) implies
±
(xS y )>, (xjy
if
(3)
1
,
x
x
goods smaller than w
and p
,
).
of section II)
so that
if
(x
i -i
p
x
*x
1
1
y
;
*w
.
y
point with all private
a
with smaller
there is a point in X
0,
>_
i
This will lead to a contradiction (as in the theorem
value than (x;y
(4)
= p
Since there is in X
Suppose equality held.
,
th en
)
-1 ^
i
i
+p-y
r
J
—>pr x "x+p-y
y
y
i ,-i
p
1
)^
(xry
.
1
then
)
+p«y-i >p x i "x+p'y i
,
= p
y
y
«w
i
.
y
This establishes the theorem.
The Core
V.
A.
Definitions
An allocation
(x;
y
,
or coalition of consumers
)>
and (x; y
.
(x;y
1
)
...,y
)
is said to be blocked by a subset
if there exists
S
for all i
e
S
(x;
y
such that [x;
,...,y
.
E
led
(y
)
with x
>_
-
w
Y.
corresponding to consumers not in
Notice that the y
S
)]
e
0,
may be zero.
An allocation is in the core of the economy if it cannot be
If an allocation is in the core, no coalition
blocked by any coalition.
can do better for all its members with its own resources.
VI.
iW
Theorem
1
:
A Lindahl Equilibrium is in the Core
If (x; y
,
.
.
.
,y
p
;
p_
x
x
;
p
y
)
is a Lindahl equilibrium
it is in the core.
Proof
(x;
y
Suppose the coalition
:
, .
.
.
,y ).
could block (x; y ,...,y
S
Since (x;y )r
.
(x;,y
)
for all i
e
S, we
)
by
know by the
-10-
definition of Lindahl equilibrium that
.E
(1)
Since pr
(2)
+
p
•
.Z
>
y
*y ±eS J
n
i
x
'x
p
ieSc r x
>
—
for all i,
.
.
i
—>
Z..p
1=1 x
Z„p
ieS x
.
1
JiP/'X + Py-ils^-w )
+
*x
Z,.p
ieS r x
>
i
,
= n
'.Z
7
*y leSc y
p
f
«.E„w
y ieS
.
which means that, since x
'
—>
0,'
0.
But the profit maximizing condition for Lindahl equilibrium asserts that
(3)
n
.Z,p
r
i=l x
*
_
-x
_
+
p
r
-z
y
for all (x;z)
—<
e
Y.
This contradiction establishes the theorem.
VII.
Conclusion
There is no reason to think that Lindahl equilibrium can be
embodied by any working political process.
There is some reason to think
that the core is a meaningful political concept.
If a group of people
find themselves able, using only their own resources, to achieve a better
life, it is not unreasonable to suppose that they will try to enforce this
threat against the rest of the community.
They may find themselves frustrated
if the rest of the community resorts to violence or force to prevent them
from withdrawing.
In the absence of some assumption about the reaction of
the part of the society not in the coalition to the coalition threat,
detailed predictions of the equilibrium situation are not possible.
But if a society stays inside the core as it is defined here, there
is a minimal rationale for everyone to continue to participate.
The
conflicts that will naturally arise over the redistribution of initial endow-
ments will still be there.
But no group will have the power to alter the situation
in its own favor unilaterally.
This is a very crude and intuitive argument for
the relevance of the core to political reality, but the whole theory of political
equilibrium is in its infancy.
Duncan
K.
Foley
Massachusetts Institute of Technology
Bibliography
[1]
[2]
Theory of Value
Debreu, Gerard.
Sons, 1959.
.
New York: John Wiley
&
"New Concepts and Techniques for Equilibrium
Analysis," International Economic Review 3 (September, 1962),
pp. 257-273.
.
,
[3]
"Resource Allocation and the Public Sector," Yale
Economic Essays 7 (Spring, 1967), pp. 43-98.
Foley, D.
,
[4]
Johansen, Leif.
"Some Notes on the Lindahl Theory of
Determination of Public Expenditures," International Economic
Review 4 (September, 1963), pp. 346-358.
,
[5]
Karlin, S. Mathematical Methods and Theory in Games, Programming
and Economics
Reading, Mass.: Addison-Wesley 1959.
[6]
,
,
.
Lindahl, Erik.
"Just Taxation - A Positive Solution," reprinted
in part in Richard A. Musgrave and Alan T. Peacock (eds.)
Classics in the Theory of Public Finance
London: Macmillan, 1958.
.
[7]
Samuelson, P. A.
"The Pure Theory of Public Expenditures,"
Review of Economics and Statistics 36 (1954), pp. 387-389.
,
[8]
[9]
"Pure Theory of Public Expenditure and Taxation,"
unpublished manuscript.
.
"A Limit Theorem on the Core of an
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pp. 235-246.
Scarf, H. and Debreu, G.
,
Date Due
IBL
20 "If
Sep g
AU6
87*
4 'r%
-«.
AUG
2 3 198!
i
Lib -2 6 -6 7
MIT LIBRARIES
15A 771
TOflO D03
3
MIT LIBRARIES
3
"nQfiD
DD3 T27 L51
3
TDflO
DD3 T5S bbM
MIT LIBRARIES
3
TDfiD
DD3
T5fi
714
MIT LIBRARIES
P!l
3
TDflD
DD3
TSfl
bE3
MIT LIBRARIES
3
TDflD
DD3
=156
7fc,3
MIT LIBRARIES
3
TDflD
DD3 TE? hT3
MIT LIBRARIES
3
TDflO
DD3
TSfl
3
TDflD
A3T
DUPL
MIT LIBRARIES
DD3 TE7 bbT
MIT LIBRARIES
3
TDflO
DD3
=127
b77
M
G"
i.G
*fSH