The Menu-Induced Core of an Economy with an Excludable Public Good ()
1. Introduction
This paper examines economies with excludable public goods. The public goods that we consider are those that admit partial exclusion: the amount that each agent consumes may vary from agent to agent. Tollways, pay-per-view TV programs, and public transportation are traditional examples. More recent examples include online commodities such as music and movie downloading services and access rights to databases through the internet. The early researches on such commodities are, for example, Oakland [1] and Drèze [2] .
We consider stable allocations in cooperative decision situation of such an economy. In particular, we consider the core of the coalitional form of the economy. If we allow each coalition to achieve allocations with the individualized lump-sum payments, then the core essentially turns out to be that of Foley [3] . On the other hand, many examples of the excludable public goods are allocated through prices that may be nonlinear. For example, toll fare for the highway and a bundling sale of pay-per-view TV programs. According to this practice, we allow each coalition to achieve allocations via menus, which are a kind of nonlinear price.
A nonlinear price is a kind of system. In the literature, several authors considered the core of economies where each coalition is allowed to achieve an allocation via a given system. Guesnerie and Oddou [4] [5] considered the core of a pure public good economy where each coalition achieves an allocation via proportional income tax. Spulber [6] considered the core of a production economy with increasing returns to scale where each coalition achieves an allocation via the average cost price. Hara [7] [8] considered the exchange economies where allocations are achieved via menus. Indeed, our concept of the coalitional improvement and the core is an application of his menu-induced improvement and the anonymous core to the economy with an excludable public good.
To justify such a way to achieve an allocation, we implicitly assume the same informational constraint as Hara [7] [8] . He assumed that each agent knows the distribution of the characteristics of the agents, but cannot identify who has which characteristic. Under such an informational constraint, it is difficult for the agent to find appropriate members of a coalition and an appropriate allocation achieved in the coalition. In this situation, employing a menu for achieving an allocation is legitimate for revealing the characteristics of the agents.
Our core concept is defined as the menu-induced core. It is defined as the set of allocations satisfying the following two conditions: the allocation is achievable within the grand coalition via a menu; and no coalition can achieve an allocation via a menu that makes each agent in the coalition better off. We show the nonemptiness of the menu-induced core and observe some properties, which clarify the difference between the menu-induced core and the standard Foley’s core.
In the next section, we introduce the model of the economy with an excludable public good. In Section 3, we define the menu-induced core and prove its nonemptiness. Then, we investigate certain properties of the menu-induced core in Section 4. In the final section, we conclude with some remarks.
2. The Economy with an Excludable Public Good
We consider an economy consisting of 
agents, one private good, and one public good that is partially excludable. The set of agents is denoted by
. A nonempty subset 
of 
is called a coalition. The set of all coalitions is denoted by
.
A typical consumption of 
is denoted by
, where 
and 
denote the amounts of the private good and the public good that 
consumes, respectively. The preferences of each 
are represented by a utility function
. For each
, it is assumed that 
is continuous and monotone (i.e., 
implies
). Each 
is endowed with 
of the private good.
Each coalition can access to an identical production technology that transforms the private good into the public good. The production technology is represented by a cost function
. It is assumed that 
is continuous, monotone (i.e., 
implies
), 
, and
.
For each
, an 
-allocation is a tuple of the consumptions of agents in
. For each
, an 
allocation 
is said to be 
-feasible iff
. Let 
be the set of 
feasible allocations for each
. An 
-allocation and an 
-feasible allocation are simply called an allocation and a feasible allocation, respectively. Given an allocation
, an 
-allocation 
is said to be an improvement upon 
for 
iff 
and 
for all
.
Definition 1. An allocation 
is in the standard core iff 
and there exists no 
and 
that is an improvement upon
.
Obviously, the standard core is essentially equivalent to that of Foley [3] for economies with pure public goods.
3. The Menu-Induced Core
This section introduces the main concepts of this paper, which are defined as an application of the similar concepts of Hara [7] [8] with slight modifications. Then, its nonemptiness is proved.
The menu is defined as a subset 
of
. It is a set of net consumptions: 
means that 
of the private good must be paid for consuming 
of the public good. Thus, the resulting consumption of 
choosing 
is
.
For each
, let 
be a set of 
-allocations such that 
iff 
and there exists some 
such that for all
, 1) 
and 2) 
implies
. A tuple 
is called a menu-induced coalitional form of an economy.
An allocation 
is said to be menu-induced iff
. This definition describes the situation as follows: once a menu is proposed, each agent chooses the most preferable net consumption from the proposed menu. The menu can be proposed by an agent who may find it by oneself or consulting with an outside intervener.
Given an allocation
, a coalition 
has a menu-induced improvement upon 
iff there exists some 
such that 
for all
. The improvement by a menu describes the following situation. Given an allocation, an agent proposes a menu to agents in the economy. Again, the proposing agent may find it by oneself or consulting with an outside intervener. Then, each agent responds to the menu when one can make oneself better off by choosing the most preferable net consumption from the menu. The coalition is formed among the agents who respond to the proposed menu. One may notice that the definition of the menu-induced improvement does not take the behavior of the agents outside the coalition into account. The original definition of Hara [7] [8] described the behavior of the agents outside the coalition as follows: no agent outside the coalition finds it more preferable to choose a net consumption from the menu. Thus, we may also impose the following condition on the menu-induced improvement 
upon a given allocation 
for
: for all
, 
for any
.
This condition is, however, redundant in our economy from the nonrivalry of the excludable public good. More precisely, for any allocation, there is a menu-induced improvement upon the allocation for 
if and only if there is a 
-allocation with 
that satisfies both the definition of the menu-induced improvement and the additional condition. Therefore, imposing the additional condition does not change the nature of the menu-induced core that is defined below.
Definition 2. An allocation 
is said to be in the menu-induced core iff 
and there exist no 
and 
such that 
is a menu-induced improvement upon 
for
.
Now, we prove the nonemptiness of the menu-induced core.
Theorem 1. The menu-induced core is nonempty.
Proof. For any
, define
Further, define
Clearly, 
since
. It can be easily confirmed that both 
and 
are compact sets by the assumptions of 
and
.
Define 
for each
. Note that 
is a nonempty interval of 
for any 
by definition.
First, assume that there exists some 
such that 
is unbounded above. Then, by the monotonicity of
,
(1)
for all 
and any 
with
. We claim that 
is in the menu-induced core.
Suppose that there exist some 
and a menu-induced improvement 
upon 
for
. Then, 
for all
. By (1), it must be 
for all
. It follows that
from the 
-feasibility of
, a contradiction. Hence 
is in the menu-induced core.
Next, assume that 
is bounded for any
. We claim that 
is closed for any
. Fix an arbitrary
. Let 
be a sequence taken from 
that converges to
. Then, we have 
for all 
and for all
. It follows from the continuity of 
that 
for all
. Thus, in conjunction with the boundedness, 
is a compact set for any
.
Thus, we can define 
for any
. We claim that 
is an upper semi-continuous function. It suffices to show that 
is closed for any
by the definition of
. Fix an arbitrary
. Let 
be a sequence taken from 
that converges to
. Then, we have 
for all 
and for all
. It follows from the continuity of 
that 
for all
. Thus,
. Hence 
is an upper semi-continuous function.
Since 
is a compact set and 
is an upper semi-continuous function, there exists some 
such that 
for any
. We claim that 
is in the menuinduced core.
Let
. Suppose that there exist some 
and a menu-induced improvement 
upon 
for
. By definition, 
for all
. Define
.
Then,
. Thus, 
is still a menuinduced improvement upon 
for 
by its construction.
Then, 
for all 
by the monotonicity of
. Define
where 
is a sufficiently small real number so that 
for all
. Define
where 
for each
. Note that 
for all 
since 
for all 
by its construction.
We confirm that
. The feasibility follows from 
By the construction of
, 
for all
. By the definitions of 
and
, 
for any 
and all
. Hence
. Moreover, 
since
. Thus,
. This contradicts the definition of
. Hence 
is in the menu-induced core. 
Note that we require neither the quasi-concavity of the utility functions nor the convexity of the cost function.
Note also that the proof of Theorem 1 applies the idea of Mas-Colell [9] , which provides an alternative proof of a result of Champsaur [10] : the core of the coalitional game derived from an economy with one private good and one pure public good is nonempty. One may, therefore, consider that Theorem 1 is not surprising. However, the menu-induced core is generally much different from the standard core, which will be discussed in the next section.
4. Observations
In this section, we observe some properties of the menu-induced core, and discuss the difference from the standard core. One straightforward property from the definition is the symmetry of the allocations. Two agents 
and 
are symmetric iff 
for any 
and
. An allocation 
is symmetric iff 
for any symmetric agents 
and
. It can be easily confirmed that any allocation in the standard core may not be symmetric.
Property 1. Any allocation in the menu-induced core is symmetric.
The next property shows that the coalitional form of our economy may not satisfy the following usual property. The coalitional form of an economy 
is said to be superadditive at 
iff for all disjoint
, for any 
and any
, there exists some 
such that 
for all 
and 
for all
.
Property 2. 
may not be superadditive at
.
The following example shows Property 2.
Example 1. Let 
. Let 
for all 
and
, where 
is sufficiently large. Let 
for all 
and
. Clearly, 
and
. We have
, 
for all 
and 
.
Let
, induced by a menu
, such that 
and
. Then, 
for some 
and some
. Then, it must be 
for each
, 
and
. Thus,
(2)
Thus, the most LHS of (2) is negative since 
is sufficiently large. In this case, 
is not feasible. Hence the coalitional form of the economy may not be superadditive. 
Property 2 shows that our coalitional form is quite different from those in the literature. The coalitional form of an economy with one pure public good and one private good is known to be ordinary convex (see for example Peleg [11] for the definition of the ordinary convexity), which is a stronger condition than the superadditivity. See also Ichiishi [12] . On the other hand, Guesnerie and Oddou [4] [5] considered a coalitional form of an economy with one pure public good and one private good where each coalition is allowed to achieve an allocation through a proportional income tax. They showed that this coalitional form may not be superadditive and the core may be empty, while the menu-induced core is always nonempty in spite of the nonsuperadditivity.
The next property shows that there is a case where the menu-induced core is a subset of the standard core.
Property 3. If all agents are symmetric, then the menu-induced core is included in the standard core.
Proof. Assume that all agents are symmetric. Fix an arbitrary 
that is in the menu-induced core. For each
, define 
for any
.
We claim that 
for all
. Suppose that there exists some 
such that
. Then, by the symmetry of the agents, 
for all
. Let 
for all
. Note that such 
exists by the symmetry of the agents. Then, we can easily confirm that 
where 
and 
for all 
is a menu-induced improvement upon
. This contradicts that 
is in the menu-induced core. Hence 
for all
.
Then, we show that 
is in the standard core. Suppose that there exist some 
and an 
- feasible allocation 
that is an improvement upon 
for
. Let
. Since 
for all
, 
for all
. Thus, 
for all
. Then, 
, contradicting that
. Hecne 
is in the standard core. 
On the other hand, there is a case where the menu-induced core is disjoint with the standard core.
Property 4. The intersection of the menu-induced core and the standard core may be empty.
The following example shows Property 4, which is a modification of an example in Guesnerie and Oddou [5] .
Example 2. Let
, 
for 
and 
for 
, where
, and 
for all
. The cost function is represented by
.
At any allocation 
in the standard core, it can be easily confirmed that 
for all 
since the utility functions are continuous and strictly increasing in the interior of
.
Let 
be an allocation in the standard core. If 
is a menu-induced allocation, then it must be 
for all 
since 
and 
for all
. Thus, there exists some 
such that 
for all
.
Let us consider two menus 
and
. If at least 
agents prefer 
(
, respectively) to
, then 
is not in the menu-induced core. Thus, if 
is in the menu-induced core, it must be both
(3)
(4)
However, there exists no 
that simultaneously satisfies (3) and (4) if 
is sufficiently small. Thus, in this example, the intersection of the standard core and the menu-induced core is empty. 
By Property 3 and 4, there is no general relationship between the menu-induced core and the standard core.
5. Concluding Remarks
This paper defined the menu-induced core and showed its nonemptiness in an economy with an excludable public good. We also discussed some properties of the menu-induced core. One remaining problem is the evaluation of the efficiency of the menu-induced core. In general, the menu-induced core fails to achieve the Pareto efficiency, which may be caused by the underlying informational constraints. We may consider the extent of the inefficiency, or efficiency evaluation under the similar informational constraints.
Another remaining problem is to design a mechanism that implements the allocations in the menu-induced core. In the economy with an excludable public good, some mechanisms are proposed. For example, Moulin [13] proposed the serial cost sharing mechanism, and Moldovanu [14] and Bag and Winter [15] proposed mechanisms that implement the standard core. However, all of them do not implement the menu-induced core. We leave these problems for future research.
Acknowledgements
I am grateful to an anonymous reviewer and Mikio Nakayama for their helpful comments and advices. I am also grateful for the financial supports by JSPS Grant-in-aid for Young Scientists (B) 22730155 and JSPS Grant-in-aid for Scientific Research (B) 24310110.