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Proportional income taxes and cores in a public goods economy
JOURNAL
OF
ECONOMIC
THEORY
15, 295-300 (1977)
Proportional
Income Taxes
and Cores in a Public Goods Economy MIKIO Faculty
of Ecmotnics,
Toyama
NAKAYAMA
University,
3190 Gofiku,
Toyama
930, Japan
Received October 4, 1976: revised January 3, 1977
An equivalence relationship between cores and Pareto optimal allocations of a public goods economy where the public goods are to be provided through a proportional income tax is presented. For this purpose, the definition of the core is modified by allowing coalitions to tax their complements at any given rate. Also, a certain rule which specifies the rate is introduced.
The relationships between equilibria and the core of economies with public goods have been investigated well in recent years. In particular, Muench [6], Champsaur, Roberts, and Rosenthal [2], and Champsaur [I] indicate that equivalence theorems between Lindahl equilibria and the core would be difficult to obtain. More recently, however, Kaneko [4] has presented a new equilibrium concept called a ratio equilibrium, and has shown that the ratio equilibria coincide with the core of a certain voting game. The purpose of this article is to examine similar relationships in a more familiar framework-when public goods are provided through a proportional income tax. Resource allocation through a tax system in an economy with public goods was defined by Foley [3] as a public competitive equilibrium. He also showed elsewhereits existence when a tax takes the form of a proportional income tax. The core we consider is defined under a modification that coalitions are endowed with the power to tax their complements in making new proposals. Thus the core is related to a tax rate f at which every individual would have to sacrifice his income for public goods under a proportional income tax system. We call it a t-core. The t-core may allow us to consider some government intervention to narrow the core by manipulating the level of the rate t in an analogous way to that mentioned in Nakayama [7]. The definition of the t-core may follow primarily from the concept of Y-cores defined by Champsaur, Roberts, and Rosenthal [2], sinceit embodies the notion that coalitions are allowed certain powers to enforce contributions Copyright All rights
0 1977 by Academic Press, of reproduction in nny form
Inc. reserved.
295 ISSK
0022-0531
296
MlKlO
NAKAYAMA
upon their complements in producing public goods. But we require one more condition to offset the increased power of coalitions, that new allocations proposed against a current one must contain at least an equal amount of each public good. This additional constraint makes the t-core much different from Y-cores or the usual core. Our first result is a characterization of the ?-core for any given rate t. Then we introduce a certain rule which specifies the rate t in making a new proposal, and thereby show some equivalence between the t-core and the allocation achieved under the tax rule.
t-ALLOCATIONS
We assume that our economy consists of n individuals, m public goods, and money. The set of all individuals is denoted by N = (I,...., n}. Each individual i E N initially posseses Mi (>O) amount of money only. The utility of i E N in consuming qi (20) amount of each public good j and xi (30) amount of money is given by ui (q, x,), where q == (qI ,..., qm). The cost needed to produce qj amount of the public good j is given by Cj(qj), which is measured in terms of money. We assume that for all i E N, ui(q, xi) is continuous and strictly monotone - increasing, i.e., 2jj > qj for allj, Xi 5Y xi and (4, Xi) f (4, xi) imply u&, 4 > ui(q, xi), and assume that Cj(qj) is continuous, monotone increasing, and satisfies C,(O) = 0 for allj = l,..., m. Let us write x(S), M(S) for CisS xi , CisS Mi , respectively, for any nonempty subset S of N, and C(q) for Cy=, Cj(qj). A feasible allocation in this economy is the pair (q, x) = (ql ,..., qvP1, x1 ,..., x,) satisfying for all j, i and
C/j > O, Xi >, O A t-allocation O
is the feasible allocation and
C(q) = twm
x(N) $- C(q) < M(N).
(q, x) such that xi = (1 - t)Mi
A Pareto optimal allocation is the feasible allocation - Ui(q,
xi)
>
ui(45
xi)
for all
for all
i E N.
(q, x) for which ieN
implies x(N)
+ c(q) > M(N).
A t-optimal allocation is the t-allocation (q, x) which is Pareto When (q, x) is t-optimal, we say t is an optimal tas rate.
optimal.
297
TAXES AND CORES ~-CORES
Let a tax rate t be given. Once this taxation is carried out by public authorities, t-allocations will result without any reference to individual welfare except possibly Pareto optimality. Suppose alternatively that any coalition of individuals is allowed to collect taxes at the same rate t from its complementary coalition in order to propose a new allocation. Then, we say an allocation (q, x) is t-dominated by (4, F) if there is a nonempty subset S of N such that
&(q, 3 > 4(q, Xi)
for all
X(S) + c(q) < M(S) + ?M(N - S) if
S f N,
then qj 3 qj
ieS
(1)
(O,(t
(2)
for all j = I,..., m.
(3)
Conditions (1) and (2) imply that the coalition must find within its budget increased by the tax revenue an allocation which is more desirable to it. Condition (3) requires that the new allocation must contain at least an equal amount of each public good, if the complementary coalition is a nonempty set. This additional constraint serves to offset the increased power of the coalition. Against t-allocations, where t is the same rate, this constraint implies that the coalition must not consume the tax revenue for its private purposes. The t-core is thendefined as the set of allocations which are not t-dominated. It immediately follows from the definition that the &-core includes the t,-core if t, ,( t, . We show initially that for any given t (0 < t ,< I) the t-core is the set of all Pareto optimal allocations in which every individual pays for the public goods at least the amount required under the tax system. PROPOSITION 1. Let K(t) be the set of a21Pareto optimal allocations (q, x) satisfying xi < (1 - t)M, for all i E N. Then the t-core equals the set K(t).
Proof. Let (q, X) E K(t). Suppose (q, x) is t-dominated by (ij, X). Then there is a proper subset S for which (q, F) satisfies (1), (2), and (3). If Xi > 0 for all i E S, we can find an allocation (q’, x’) such that
4’ = P, x,’ = xi - 6
if i E S (6 > 0 is sufficiently small) = xi + sS/(n - s) if i E N - S (s is the cardinality
of S).
This (q’, x’) is feasible by (2) and the fact that xi < (1 - t)Mi for all i E N. But continuity and monotonicity of ui imply ui(q’, xi’) > ui(q, xi) for all i E N. This contradicts that (q, x) is Pareto optimal. If Xi = 0 for some i E S,
298
MIKIO
NAKAYAMA
then we have yi > qi for some j by monotonicity feasible allocation (q”, x”) such that
of ui . Then, there is a
But, again, we have ui(q”, xi) > u,(q, xi) for all i E N, a contradiction. Conversely, let (4, X) be any feasible allocation such that (4, X) $ K(t). Assume that (4, X) is Pareto optimal. Then we can always choose a nonempty proper subset S such that X(N - S) > (1 - t) M(N Feasibility
- S).
of (4, X) then implies a(s) + c(q) < M(S) + tM(N
Hence there can be found an allocation qj'
>
qj
x.’0,)1 j-
for all
- S).
(q’, x’) whidsatisfies
j = l,..., m and q’ # 4,
for all i ES,
and x’(S) + C(q’) e M(S) + tM(N - S). This allocation (q’, x’) t-dominates (4, X). If (4, X) is not Pareto optimal, then it is t-dominated by definition. Q.E.D. Let t* be an optimal tax rate. Then the t*-optimal allocations are in the t*-core, but the t*-core includes many other allocations in general. Since the t-core shrinks monotonically as the rate t is raised, we might hope that the t-core eventually shrinks to a set of some meaningful allocations, e.g., t-allocations, for our purpose. However, this does not hold. Let t-core be the minimal nonempty t-core. Then it follows immediately that the t-core consists of those allocations (4, X) in K(t) such that Xi = (1 - t)M, for some i E N. Hence the i-core does not coincide with the set of i-allocations in general. We shall now consider an alternative way to attain t-optimal allocations without any intervention from outside of N. We introduce a rule which must be followed by any coalition in making new proposals. Let t(x) be a function defined by t(x) = 1 - x(N)/M(N)
TAXES AND CORES
299
for any feasible allocation (4, x). Note that 0 & t(x) < 1. The function t(x) describes the rate at which money is spent as a whole for the public goods in allocation (9, x). When a coalition intends to improve upon (q, x), the coalition must be able to find an allocation (4, X) which satisfies (l), (3), and (2’), where (2’) is given by X(S) + c(q) .< M(S) T t(x) M(N - S).
w
This is our new rule of domination. Note that we are considering domination in effectivenessform introduced by Rosenthal [8], since the effective set of a coalition depends on the current allocation to be improved. Kaneko [5] has applied this notion to the core of a certain voting game to show the equivalence of the core to ratio equilibria. We show an analogous relationship between t-cores and t-optimal allocations. Consider the set of allocations which are undominated in the sensedefined above. We call the set an e-core. Then an allocation (q, x) belongs to the e-core if and only if it belongs to the t-core, where t = t(x). PROPOSITION 2. The e-core coincides with the set of those allocations (q, s) trhich are t(x)-optimal.
Proof. Let ((1.x) be t(x)-optimal. Then, letting t = t(x), we know from Proposition 1 that (q, x) is not t-dominated. Hence (q, x) is in the e-core. Conversely, suppose (q, x) is not t(x)-optimal. If (q, x) is not Pareto optimal, then it is clearly not in the e-core. If (q, x) is not a t(x)-allocation, then there is an i EN such that xi > (1 - t(x))M, . Then, Proposition 1 implies that (q, x) is t-dominated, where t = t(x). Hence (q, x) is not in the c-core. Q.E.D.
CONCLUDING
REMARKS
The t-core we have considered is much different from the usual core. For instance, the O-core is the set of all Pareto optimal allocations, while the l-core is empty unless the l-allocation (ql ,..., q,n, O,..., 0) is Pareto optimal. This difference is due to condition (3) in the absence of which nonempty t-cores would be included in the usual core. It should also be noted that there may be allocations (q, x) in the t-core, and hence in the e-core, such that the condition ui(q, xi) > ~~(0, M.+) for all i E N is not satisfied. If we consider that individuals are free to leave this public goods economy, individual rationality of this kind might be needed. In this casethis condition can be added to the definition of the t-core.
300
MIKIO
NAKAYAMA
REFERENCES 1.
P.
How to share the cost of a public good ? Izzterrzat.
CHAMPSAUR,
4 (1975),
J. Game
Theory
113-129.
P. CHAMPSAUR, D. J. ROBERTS, AND R. W. ROSENTHAL, On cores in economies with public goods, Znterrzat. Econ. Rev. 16 (1975), 751-764. 3. D. FOLEY, Lindahl’s solution and the core of an economy with public goods, Econornrt2.
rim
38 (1970),
M.
KANEKO,
66-72.
The ratio equilibrium and a voting game in a public goods economy, J. Econ. Tlleory, to appear. 5. M. KANEKO. “The Ratio Equilibria and the Core of the Voting Game G(N. W) in a Public Goods Economy,” Research Reports on Information Sciences No. B-31, Department of Information Sciences, Tokyo Institute of Technology, June 1976. 6. T. MUENCH, The core and the Lindahl equilibrium of an economy with a public good: An example, J. Econ. Theory 4 (1972), 241-255. 7. M. NAICAYAMA, A note on the core of an income redistribution game, J. Ecorr. Theory 4.
12 (1976), 8. R.
W.
88-101.
347-349.
ROSENTHAL,
Cooperative games in effectiveness form, J. Econ.
Theory
5
(1972),
OF
ECONOMIC
THEORY
15, 295-300 (1977)
Proportional
Income Taxes
and Cores in a Public Goods Economy MIKIO Faculty
of Ecmotnics,
Toyama
NAKAYAMA
University,
3190 Gofiku,
Toyama
930, Japan
Received October 4, 1976: revised January 3, 1977
An equivalence relationship between cores and Pareto optimal allocations of a public goods economy where the public goods are to be provided through a proportional income tax is presented. For this purpose, the definition of the core is modified by allowing coalitions to tax their complements at any given rate. Also, a certain rule which specifies the rate is introduced.
The relationships between equilibria and the core of economies with public goods have been investigated well in recent years. In particular, Muench [6], Champsaur, Roberts, and Rosenthal [2], and Champsaur [I] indicate that equivalence theorems between Lindahl equilibria and the core would be difficult to obtain. More recently, however, Kaneko [4] has presented a new equilibrium concept called a ratio equilibrium, and has shown that the ratio equilibria coincide with the core of a certain voting game. The purpose of this article is to examine similar relationships in a more familiar framework-when public goods are provided through a proportional income tax. Resource allocation through a tax system in an economy with public goods was defined by Foley [3] as a public competitive equilibrium. He also showed elsewhereits existence when a tax takes the form of a proportional income tax. The core we consider is defined under a modification that coalitions are endowed with the power to tax their complements in making new proposals. Thus the core is related to a tax rate f at which every individual would have to sacrifice his income for public goods under a proportional income tax system. We call it a t-core. The t-core may allow us to consider some government intervention to narrow the core by manipulating the level of the rate t in an analogous way to that mentioned in Nakayama [7]. The definition of the t-core may follow primarily from the concept of Y-cores defined by Champsaur, Roberts, and Rosenthal [2], sinceit embodies the notion that coalitions are allowed certain powers to enforce contributions Copyright All rights
0 1977 by Academic Press, of reproduction in nny form
Inc. reserved.
295 ISSK
0022-0531
296
MlKlO
NAKAYAMA
upon their complements in producing public goods. But we require one more condition to offset the increased power of coalitions, that new allocations proposed against a current one must contain at least an equal amount of each public good. This additional constraint makes the t-core much different from Y-cores or the usual core. Our first result is a characterization of the ?-core for any given rate t. Then we introduce a certain rule which specifies the rate t in making a new proposal, and thereby show some equivalence between the t-core and the allocation achieved under the tax rule.
t-ALLOCATIONS
We assume that our economy consists of n individuals, m public goods, and money. The set of all individuals is denoted by N = (I,...., n}. Each individual i E N initially posseses Mi (>O) amount of money only. The utility of i E N in consuming qi (20) amount of each public good j and xi (30) amount of money is given by ui (q, x,), where q == (qI ,..., qm). The cost needed to produce qj amount of the public good j is given by Cj(qj), which is measured in terms of money. We assume that for all i E N, ui(q, xi) is continuous and strictly monotone - increasing, i.e., 2jj > qj for allj, Xi 5Y xi and (4, Xi) f (4, xi) imply u&, 4 > ui(q, xi), and assume that Cj(qj) is continuous, monotone increasing, and satisfies C,(O) = 0 for allj = l,..., m. Let us write x(S), M(S) for CisS xi , CisS Mi , respectively, for any nonempty subset S of N, and C(q) for Cy=, Cj(qj). A feasible allocation in this economy is the pair (q, x) = (ql ,..., qvP1, x1 ,..., x,) satisfying for all j, i and
C/j > O, Xi >, O A t-allocation O
is the feasible allocation and
C(q) = twm
x(N) $- C(q) < M(N).
(q, x) such that xi = (1 - t)Mi
A Pareto optimal allocation is the feasible allocation - Ui(q,
xi)
>
ui(45
xi)
for all
for all
i E N.
(q, x) for which ieN
implies x(N)
+ c(q) > M(N).
A t-optimal allocation is the t-allocation (q, x) which is Pareto When (q, x) is t-optimal, we say t is an optimal tas rate.
optimal.
297
TAXES AND CORES ~-CORES
Let a tax rate t be given. Once this taxation is carried out by public authorities, t-allocations will result without any reference to individual welfare except possibly Pareto optimality. Suppose alternatively that any coalition of individuals is allowed to collect taxes at the same rate t from its complementary coalition in order to propose a new allocation. Then, we say an allocation (q, x) is t-dominated by (4, F) if there is a nonempty subset S of N such that
&(q, 3 > 4(q, Xi)
for all
X(S) + c(q) < M(S) + ?M(N - S) if
S f N,
then qj 3 qj
ieS
(1)
(O,(t
(2)
for all j = I,..., m.
(3)
Conditions (1) and (2) imply that the coalition must find within its budget increased by the tax revenue an allocation which is more desirable to it. Condition (3) requires that the new allocation must contain at least an equal amount of each public good, if the complementary coalition is a nonempty set. This additional constraint serves to offset the increased power of the coalition. Against t-allocations, where t is the same rate, this constraint implies that the coalition must not consume the tax revenue for its private purposes. The t-core is thendefined as the set of allocations which are not t-dominated. It immediately follows from the definition that the &-core includes the t,-core if t, ,( t, . We show initially that for any given t (0 < t ,< I) the t-core is the set of all Pareto optimal allocations in which every individual pays for the public goods at least the amount required under the tax system. PROPOSITION 1. Let K(t) be the set of a21Pareto optimal allocations (q, x) satisfying xi < (1 - t)M, for all i E N. Then the t-core equals the set K(t).
Proof. Let (q, X) E K(t). Suppose (q, x) is t-dominated by (ij, X). Then there is a proper subset S for which (q, F) satisfies (1), (2), and (3). If Xi > 0 for all i E S, we can find an allocation (q’, x’) such that
4’ = P, x,’ = xi - 6
if i E S (6 > 0 is sufficiently small) = xi + sS/(n - s) if i E N - S (s is the cardinality
of S).
This (q’, x’) is feasible by (2) and the fact that xi < (1 - t)Mi for all i E N. But continuity and monotonicity of ui imply ui(q’, xi’) > ui(q, xi) for all i E N. This contradicts that (q, x) is Pareto optimal. If Xi = 0 for some i E S,
298
MIKIO
NAKAYAMA
then we have yi > qi for some j by monotonicity feasible allocation (q”, x”) such that
of ui . Then, there is a
But, again, we have ui(q”, xi) > u,(q, xi) for all i E N, a contradiction. Conversely, let (4, X) be any feasible allocation such that (4, X) $ K(t). Assume that (4, X) is Pareto optimal. Then we can always choose a nonempty proper subset S such that X(N - S) > (1 - t) M(N Feasibility
- S).
of (4, X) then implies a(s) + c(q) < M(S) + tM(N
Hence there can be found an allocation qj'
>
qj
x.’0,)1 j-
for all
- S).
(q’, x’) whidsatisfies
j = l,..., m and q’ # 4,
for all i ES,
and x’(S) + C(q’) e M(S) + tM(N - S). This allocation (q’, x’) t-dominates (4, X). If (4, X) is not Pareto optimal, then it is t-dominated by definition. Q.E.D. Let t* be an optimal tax rate. Then the t*-optimal allocations are in the t*-core, but the t*-core includes many other allocations in general. Since the t-core shrinks monotonically as the rate t is raised, we might hope that the t-core eventually shrinks to a set of some meaningful allocations, e.g., t-allocations, for our purpose. However, this does not hold. Let t-core be the minimal nonempty t-core. Then it follows immediately that the t-core consists of those allocations (4, X) in K(t) such that Xi = (1 - t)M, for some i E N. Hence the i-core does not coincide with the set of i-allocations in general. We shall now consider an alternative way to attain t-optimal allocations without any intervention from outside of N. We introduce a rule which must be followed by any coalition in making new proposals. Let t(x) be a function defined by t(x) = 1 - x(N)/M(N)
TAXES AND CORES
299
for any feasible allocation (4, x). Note that 0 & t(x) < 1. The function t(x) describes the rate at which money is spent as a whole for the public goods in allocation (9, x). When a coalition intends to improve upon (q, x), the coalition must be able to find an allocation (4, X) which satisfies (l), (3), and (2’), where (2’) is given by X(S) + c(q) .< M(S) T t(x) M(N - S).
w
This is our new rule of domination. Note that we are considering domination in effectivenessform introduced by Rosenthal [8], since the effective set of a coalition depends on the current allocation to be improved. Kaneko [5] has applied this notion to the core of a certain voting game to show the equivalence of the core to ratio equilibria. We show an analogous relationship between t-cores and t-optimal allocations. Consider the set of allocations which are undominated in the sensedefined above. We call the set an e-core. Then an allocation (q, x) belongs to the e-core if and only if it belongs to the t-core, where t = t(x). PROPOSITION 2. The e-core coincides with the set of those allocations (q, s) trhich are t(x)-optimal.
Proof. Let ((1.x) be t(x)-optimal. Then, letting t = t(x), we know from Proposition 1 that (q, x) is not t-dominated. Hence (q, x) is in the e-core. Conversely, suppose (q, x) is not t(x)-optimal. If (q, x) is not Pareto optimal, then it is clearly not in the e-core. If (q, x) is not a t(x)-allocation, then there is an i EN such that xi > (1 - t(x))M, . Then, Proposition 1 implies that (q, x) is t-dominated, where t = t(x). Hence (q, x) is not in the c-core. Q.E.D.
CONCLUDING
REMARKS
The t-core we have considered is much different from the usual core. For instance, the O-core is the set of all Pareto optimal allocations, while the l-core is empty unless the l-allocation (ql ,..., q,n, O,..., 0) is Pareto optimal. This difference is due to condition (3) in the absence of which nonempty t-cores would be included in the usual core. It should also be noted that there may be allocations (q, x) in the t-core, and hence in the e-core, such that the condition ui(q, xi) > ~~(0, M.+) for all i E N is not satisfied. If we consider that individuals are free to leave this public goods economy, individual rationality of this kind might be needed. In this casethis condition can be added to the definition of the t-core.
300
MIKIO
NAKAYAMA
REFERENCES 1.
P.
How to share the cost of a public good ? Izzterrzat.
CHAMPSAUR,
4 (1975),
J. Game
Theory
113-129.
P. CHAMPSAUR, D. J. ROBERTS, AND R. W. ROSENTHAL, On cores in economies with public goods, Znterrzat. Econ. Rev. 16 (1975), 751-764. 3. D. FOLEY, Lindahl’s solution and the core of an economy with public goods, Econornrt2.
rim
38 (1970),
M.
KANEKO,
66-72.
The ratio equilibrium and a voting game in a public goods economy, J. Econ. Tlleory, to appear. 5. M. KANEKO. “The Ratio Equilibria and the Core of the Voting Game G(N. W) in a Public Goods Economy,” Research Reports on Information Sciences No. B-31, Department of Information Sciences, Tokyo Institute of Technology, June 1976. 6. T. MUENCH, The core and the Lindahl equilibrium of an economy with a public good: An example, J. Econ. Theory 4 (1972), 241-255. 7. M. NAICAYAMA, A note on the core of an income redistribution game, J. Ecorr. Theory 4.
12 (1976), 8. R.
W.
88-101.
347-349.
ROSENTHAL,
Cooperative games in effectiveness form, J. Econ.
Theory
5
(1972),