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This paper introduces a core concept in an economy with an excludable public good. In the economy, we assume that each coalition is allowed to achieve an allocation via a menu, a kind of a nonlinear price. Our core concept is called the menu-induced core that is defined as the set of allocations achievable by menus that are robust against all coalitional improvements achieved via menus. We show that the menu-induced core is nonempty. We also investigate certain properties of the menu-induced core that show the difference between the menu-induced core and the core defined in a standard way.

This paper introduces a core concept in an economy with an excludable public good. In the economy, we assume that each coalition is allowed to achieve an allocation via a menu, a kind of a nonlinear price. Our core concept is called the menu-induced core that is defined as the set of allocations achievable by menus that are robust against all coalitional improvements achieved via menus. We show that the menu-induced core is nonempty. We also investigate certain properties of the menu-induced core that show the difference between the menu-induced core and the core defined in a standard way.

Menu-Induced Core, Excludable Public Good

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 [

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 [

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 [

To justify such a way to achieve an allocation, we implicitly assume the same informational constraint as Hara [

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.

We consider an economy consisting of

A typical consumption of

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

For each

Definition 1. An allocation

Obviously, the standard core is essentially equivalent to that of Foley [

This section introduces the main concepts of this paper, which are defined as an application of the similar concepts of Hara [

The menu is defined as a subset

For each

implies

An allocation

Given an allocation

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

Definition 2. An allocation

Now, we prove the nonemptiness of the menu-induced core.

Theorem 1. The menu-induced core is nonempty.

Proof. For any

Further, define

Clearly,

Define

First, assume that there exists some

for all

Suppose that there exist some

Next, assume that

Thus, we can define

Since

Let

Then,

Then,

where

where

We confirm that

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 [

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

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

Property 2.

The following example shows Property 2.

Example 1. Let

Let

Thus, the most LHS of (2) is negative since

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 [

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

We claim that

Then, we show that

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 [

Example 2. Let

At any allocation

Let

Let us consider two menus

However, there exists no

By Property 3 and 4, there is no general relationship between the menu-induced core and the standard core.

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 [

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.