The Weak Hybrid Equilibria of an Exchange Economy with a Continuum of Agents and Externalities

Proof of Lemma 3.1

For any given p ∈ Δ, f ∈ D _S (p) and p ⁿ ∈ Δ with p ⁿ → p, the proof is obvious if

Otherwise, since w ( t ) ∈ i n t R + L , ∀ t ∈ T and

we have

implying that there exists n₁ > 0 such that for any n > n₁,

We next define f n ∈ L 1 S , R + L , ∀ n > n 1 , as follows:

1. if

then

1. if

then

Therefore, for any n > n₁,

that is, f ⁿ ∈ D _S  (p ⁿ ), ∀n > n₁ and {f ⁿ } converges weakly to f. This completes the proof.

Proof of Lemma 3.2

Since S strongly hybrid blocks ( f ̄ , p ̄ ) ∈ L 1 T , R + L × Δ , it follows that there exist ɛ > 0 and g S ∈ D S ( p ̄ ) such that for any g N k − S ′ ∈ X N k − S ,

Since y N k → u ( t , y N k , x − N k ) is nondecreasing for any fixed x − N k ∈ R + L × ( r 0 − 1 ) , then if we select g N k − S ′ = 0 N k − S , where 0 N k − S is the almost everywhere null function, we have that

Since {u (t, ⋅)|t ∈ T} is equi-continuous, it follows that there exists open neighborhoods O ( g S ) , O ( f ̄ ) of g S , f ̄ respectively such that for any h _S ∈ O (g _S ), any f ′ ∈ O ( f ̄ ) and any t ∈ S,

By Lemma 3.1, D _S is lower semicontinous. Then, there exists an open neighborhood O ( p ̄ ) of p ̄ such that

for any p ′ ∈ O ( p ̄ ) . Thus, for any f ′ ∈ O ( f ̄ ) and any p ′ ∈ O ( p ̄ ) , there exists h _S ∈ D _S  (p′) ∩ O (g _S ) for which

Since for any k ∈ R and any t ∈ N k , y N k → u ( t , y N k , x − N k ) is concave and nondecreasing for any fixed x − N k ∈ R + L × ( r 0 − 1 ) , it follows that for any f ′ ∈ O ( f ̄ ) and any p ′ ∈ O ( p ̄ ) , there exists h _S ∈ D _S  (p′) such that for any g N k − S ′ ∈ X N k − S and any t ∈ S,

Therefore, every point of O ( f ̄ ) × O ( p ̄ ) can be strongly hybrid blocked by S. This completes the proof.

Proof of Lemma 3.3

By Step (d) of the proof of Lemma 3.2 in Yang (2020), it is similar to obtain that u ̂ k , j is continuous and concave. Moreover, by (H.5–2), it is easy to verify that

is nondecreasing for any ( x k ′ , l l ∈ M k ′ ) k ′ ≠ k ∈ R + L × ∑ r ≠ k m r .

Proof of Lemma 3.4

We shall organize the proof by the following steps.

Step 1

Some notations.

Without loss of generality, we assume that

Then, for any k ∈ R, there exist M _k = {1, … , m _k } and { Q k , j ∈ T k | j ∈ M k } such that

and for any i ∈ {1, … , n _k }, there exists B ⊆ M _k for which

Step 2

Converting into an exchange economy with finitely many agents.

We now consider an exchange economy

with the following arguments.

1. M ̂ = { ( k , j ) | j ∈ M k , k ∈ R } , Ω ̂ = { M k | k ∈ R }

2. The space of commodities is R + L , and the space of prices is

1. For any q = ( k , j ) ∈ M ̂ , the initial endowment of q is defined by

1. For any k ∈ R and any B ⊆ M _k , the feasible allocation set of the coalition B is defined by

1. For any k ∈ R and any B ⊆ M _k , the budget set of the coalition B is defined by

1. For any q = ( k , j ) ∈ M ̂ , the utility function of q is defined by

By Lemma 3.3, the exchange economy E ̂ satisfies the conditions of Theorem 2.1. Then, there exist x r , l * l ∈ M r r ∈ R ∈ R + L × ∑ r = 1 r 0 m r and p* ∈ Δ such that

1. for any k ∈ R and any B ⊆ M _k , there exists no y k , l l ∈ B ∈ D ̂ k , B ( p * ) such that for any z k , l l ∈ M k − B ∈ X ̂ k , M k − B and any j ∈ B,

Step 3

Completing the proof.

We define f * ∈ L 1 T , R + L by

Obviously, for any k ∈ R,

By the result (a) of Step 2, we have

Furthermore, by way of contradiction, we suppose that some S_k,i can strongly hybrid block (f*, p*). Then, there exists B ⊆ M _k such that S_k,i = Q_k,B. Moreover, there exist ɛ > 0 and g S k , i ∈ D k , S k , i ( p * ) such that for any h N k − S k , i ∈ X k , N k − S k , i ,

From g S k , i ∈ D k , S k , i ( p * ) , it follows that

where y k , l ( g S k , i ) ∈ R + L , ∀ l ∈ B is defined by

Thus, y k , l ( g S k , i ) l ∈ B ∈ D ̂ k , B ( p * ) and

For any z k , M k − B = z k , l l ∈ M k − B ∈ X ̂ k , M k − B , we have

where h N k − S k , i z k , M k − B ∈ L 1 N k − S k , i , R + L is defined by

Thus, h N k − S k , i z k , M k − B ∈ X N k − S k , i and for any z k , M k − B ∈ X ̂ k , M k − B , we have that

Hence, we have that y k , l ( g S k , i ) l ∈ B ∈ D ̂ k , B ( p * ) and for any z k , l M k − B = z k , M k − B ∈ X ̂ k , M k − B and any j ∈ B,

This contradicts the fact that for any k ∈ R and any B ⊆ M _k , there exists no y k , l l ∈ B ∈ D ̂ k , B ( p * ) such that for any z k , l l ∈ M k − B ∈ X ̂ k , M k − B and any j ∈ B,

The proof is completed.

Proof of Lemma 3.5

For any f ∈ Θ, by (H.2), we have that for any t ∈ T,

Let

which is nonempty and compact in R + L . Then, Θ ⊂ L₁ (T, A), where L₁ (T, A) is the space of equivalence classes of A− valued Bochner integrable functions f: T → A. By Yannelis (), L₁ (T, A) is nonempty and weakly compact. Hence, Θ is also nonempty and weakly compact. This completes the proof.

Proof of Theorem 4.1

Let ( f * , p * ) ∈ L 1 T , R + L × Δ be a competitive equilibrium of E . We have that

for any t ∈ T, f*(t) ∈ D _t  (p*) and

Suppose that (f*, p*) is not a hybrid equilibrium of E . Then, there exist k ∈ R, S ∈ T k and g _S ∈ D _S  (p*) for which

Thus, g _S (t)∉D _t  (p*), ∀t ∈ S, implying that

We get that g _S ∉D _S  (p*). It is a contradiction. This completes the proof.

Proof of Theorem 4.2

Let ( f * , p * ) ∈ L 1 T , R + L × Δ be a hybrid equilibrium of E . Then, we have that

for any k ∈ R and any S ∈ T k , there exists no g _S ∈ D _S  (p*) for which

It yields that for any t ∈ T,

We next prove that f*(t) ∈ D _t  (p*) for any t ∈ T.

Since, for any t ∈ T, there exists a sequence x t n of R + L such that x t n → f * ( t ) and

implying that x t n ∉ D t ( p * ) , that is,

As n → + ∞, we have

Since

it follows that

Then, we must obtain that

that is, f*(t) ∈ D _t  (p*), ∀t ∈ T. This completes the proof.