Equivalence of the Aubin bargaining set and the set of competitive equilibria in a finite coalition production economy

Journal of Mathematical Economics 68 (2017) 55–61

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Journal of Mathematical Economics journal homepage: www.elsevier.com/locate/jmateco

Equivalence of the Aubin bargaining set and the set of competitive equilibria in a finite coalition production economy Jiuqiang Liu ∗ School of Management Engineering, Xi’an University of Finance and Economics, Xi’an Shaanxi, 710100, PR China Department of Mathematics, Eastern Michigan University, Ypsilanti, MI 48197, USA

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Article history: Received 27 September 2015 Received in revised form 7 November 2016 Accepted 14 November 2016 Available online 27 November 2016 Keywords: Bargaining sets Cores Competitive equilibrium Coalition production economy Exchange economy

abstract Mas-Colell (Mas-Colell, 1989) proved that the bargaining set and the set of competitive allocations coincide in an exchange economy with a continuum of traders under some standard assumptions. In the case of pure exchange economies with a finite number of traders it is well-known that the set of competitive allocations could be a strict subset of the core which can also be a strict subset of the bargaining set. In this paper, we show that the Aubin bargaining set (or fuzzy bargaining set) and the set of competitive allocations coincide in a finite coalition production economy under some standard assumptions. We also show that the (Mas-Colell) bargaining set shrinks to the set of competitive allocations in a finite coalition production economy E under some standard conditions when E is replicated. As a consequence, the existence of competitive equilibrium in a finite coalition production economy implies the nonemptiness of Aubin bargaining sets. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Bargaining sets, Cores, and equilibria are important solutions for economies. In 1954, Arrow and Debreu (1954) established the celebrated existence theorem of competitive equilibria in finite exchange economies and some special finite production economies under a set of assumptions. Since then, the existence of competitive equilibrium for different economies under various assumptions has been studied extensively in the literature, including those for continuum economies by Aumann (1966), Greenberg et al. (1979), and Hildenbrand (1970). It is well-known that in a finite exchange economy, the set of competitive allocations (Walrasian allocations) could be a proper subset of the core, and the core could be a proper subset of the bargaining set. In dealing with large markets with many individually insignificant traders, Aumann (1964) proved a remarkable result that the core and the set of competitive allocations coincide in markets (exchange economies) with a continuum of traders under some standard assumptions. In 1989, Mas-Colell (1989) proved the following well-known fact: The bargaining set and the set of competitive allocations coincide in an exchange economy with a

continuum of traders under some standard assumptions. This result is extended to coalition production economies with a continuum of traders by Liu and Zhang (2016) recently. In Hervés-Estévez and Moreno-García (2016a), Hervés-Estévez and Moreno-García proved that the Aubin bargaining set (or fuzzy bargaining set) and the set of competitive allocations coincide in a finite exchange economy under some standard assumptions. In this paper, we extend this result to finite coalition production economies and show that the Aubin bargaining set and the set of competitive allocations coincide in a finite coalition production economy under some standard assumptions. We also show that the (Mas-Colell) bargaining set shrinks to the set of competitive allocations in a finite coalition production economy E under some standard conditions when E is replicated, which extends the corresponding result for exchange economies by Hervés-Estévez and Moreno-García (2016b). As a consequence, the existence of competitive equilibria in finite coalition production economies implies the nonemptiness of Aubin bargaining sets. The paper is motivated by ideas and results from GarcíaCutrín and Hervés-Beloso (1993), Hervés-Estévez and MorenoGarcía (2016a,b), Liu and Zhang (2016), and Mas-Colell (1989). 2. The main model

∗

Correspondence to: Department of Mathematics, Eastern Michigan University, Ypsilanti, MI 48197, USA. E-mail address: [email protected]. http://dx.doi.org/10.1016/j.jmateco.2016.11.003 0304-4068/© 2016 Elsevier B.V. All rights reserved.

Throughout this paper, for any vectors x, y ∈ Rl , we write x > y to mean xi > yi for all i; x = y to mean xi = yi for all i; and x ≥ y to mean x = y but not x = y.

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J. Liu / Journal of Mathematical Economics 68 (2017) 55–61

In this section, we introduce some preliminaries for finite coalition production economies or simply economies. We first recall the following concept of an economy (see Inoue, 2013) and some necessary preliminaries from Liu and Liu (2014). For simplicity, we assume that the preference orderings are representable by real valued concave continuous utility functions, which can be used to approximate rather general preference relations according to Section 4.6 in Debreu (1959). Let N = {1, 2, . . . , n} be the set of n agents and denote by N the set of all nonempty subsets (coalitions) of N. An economy E = (Rl , (X i , ui , wi )i∈N , (Y S )S ∈N ) with n agents is a collection of the commodity space Rl , where l is the number of commodities, agents’ characteristics (X i , ui , wi )i∈N , and coalitions’ production sets (Y S )S ∈N . The triple (X i , ui , wi ) is agent i’s characteristics as a consumer: X i ⊆ Rl is his consumption set, ui : X i → R is his utility function, and wi ∈ Rl is his endowment vector. The set Y S ⊆ Rl is the production set of the firm (coalition) S for which every agent i ∈ S works and Y S consists of all production plans that can be achieved through a joint action by the members of S. We use Y = Y N for the total production possibility set of the economy. Since inputs into production appear as negative components of y ∈ Y S and outputs as positive components, we must have Y S ∩ Rl+ = {0} (impossibility of free production) for any production set Y S , where Rl+ is the nonnegative orthant of the commodity space Rl . For each S ∈ N , the set FE (S ) of S-allocations is

 FE (S ) =

(x )i∈S i

  i i S (x − w ) ∈ Y . : x ∈ X for each i ∈ S and i

i

i∈S

Definition 2.1. For an allocation x ∈ X and a price vector p ̸= 0, the couple (x, p) is a competitive equilibrium of an economy E if the profit is maximized on Y and for each i ∈ N, xi satisfies the preferences of the ith consumer under the constraint p · xi ≤ p · wi , that is, for each i ∈ N, p · xi = p · wi

and ui (vi ) > ui (xi ) imply p · vi > p · xi .

We remark that when (x, p) is a competitive equilibrium for an economy satisfying assumption  (P.1), x is an allocation implies that there exists y ∈ Y such that i∈N (xi − wi ) = y and p · y = 0. Since p · xi ≤ p · wi for each i ∈ N, we have p · xi = p · wi for every i ∈ N. The next concept of a (Mas-Colell) bargaining set is a natural extension of the one given by Mas-Colell (1989) for exchange economies. Definition 2.2. An objection to an allocation x is a pair (S , y), where S ∈ N and y is defined on S, such that i i S (a) i∈S (y − w ) ∈ Y , i i i i (b) u (y ) ≥ u (x ) for each i ∈ S with at least one of these inequalities being strict.



Definition 2.3. Let (S , y) be an objection to an allocation x. A counterobjection to (S , y) is a pair (Q , z), where Q ∈ N and z is defined on Q , such that

(zi − wi ) ∈ Y Q , (b) u (z ) > ui (yi ) for each i ∈ S ∩ Q and ui (zi ) > ui (xi ) for each i ∈ Q \ S.

The set of all allocations of the economy E is

(a)

 F (E ) = FE (N ) =

The following concept of competitive equilibrium for an economy satisfying assumption (P.1) is given in Liu and Liu (2014).

 i

(xi )i∈N : xi ∈ X i for each i ∈ N 

 (xi − wi ) ∈ Y N = Y and i∈N

which is assumed to be nonempty and compact. We make the following assumptions on consumption sets, utility functions, and the sets of allocations: (A.1) For every agent i ∈ N, X i = Rl+ and wi > 0. (A.2) Desirability (of the commodities): x ≥ y implies ui (x) > ui (y). (A.3) For each i ∈ N, ui : X i → R is continuous and strongly i i concave (i.e., for all xi and x such that xi ̸= x and ui (xi ) ≥ i i i i i u (x ), and for all α with 0 < α < 1, u (α x + (1 − α)x ) > i i u (x )).

 each S ∈ N , FE (S ) ̸= ∅ if and only if ( i∈S X i ) ∩ Note ithat for ( i∈S w + Y S ) ̸= ∅, and 0 ∈ Y S implies that (wi )i∈S ∈ FE (S ). Note that any economy E generates an NTU game VE : N → Rn by defining, for each S ∈ N , VE (S ) = {v ∈ RS : there exists (xi )i∈S ∈ FE (S ) such that vi ≤ ui (xi ) for every i ∈ S }, where RS = {x ∈ Rn : xi = 0 for each i ∈ N \ S }. We make the following assumption on the production sets: (P.1) Y S is a convex cone containing the origin as the vertex for each S ∈ N .

l

i Remark A. Denote the price set by P = {p ∈ Rl+ : i=1 p = 1}. As remarked by Debreu and Scarf (1963) and Liu and Liu (2014), for an economy under the assumption (P.1), we have that for any price vector p ∈ P at equilibrium and for any coalition S ∈ N , p · y ≤ 0 for any y ∈ Y S . It follows that the maximum profit max{p · y : y ∈ Y S } = sup{p · y : y ∈ Y S } = 0 at any price p at equilibrium on Y S for each S ∈ N .

i∈Q i

Definition 2.4. An objection (S , y) is said to be justified if there is no counterobjection to it. The (Mas-Colell) bargaining set B (E ) of a finite coalition production economy E is the set of all allocations which have no justified objection. It is well-known that in a finite exchange economy which is a special economy with Y S = {0} for all S ∈ N , the set of competitive allocations could be a proper subset of the core which can also be a proper subset of the bargaining set. In Section 4, we show that in a finite coalition production economy under some standard assumptions, the set of competitive allocations coincides with the Aubin bargaining set—a refinement of the bargaining set through Aubin’s veto mechanism. 3. The associated continuum model In this section, we refer readers to Aumann (1964) and Liu and Zhang (2016) for the concept of a continuum economy—a coalition production economy with a continuum of agents. For a continuum economy, the set of agents is the closed interval [0, 1], denoted by T . Let X : T → Rl+ be a measurable consumption correspondence, where X (t ) is interpreted as the consumption set of agent t ∈ T . Given an economy E = (RL , (X i , ui , wi )i∈N , (Y S )S ∈N ) with n agents satisfying assumption (P.1), we construct a special continuum economy EC with n types of distinct agents as follows: We divide the set T = [0, 1] of agents into n subintervals Ii = [ i−n1 , ni ) for 1 ≤ i ≤ n − 1 and In = [ n−n 1 , 1], where all agents in Ii are identical to agent i in the economy E , that is,

EC = (Rl , X , (≻t , w(t ))t ∈T , (Y S )S ∈F , β(t , p)t ∈T ,p∈P ),

J. Liu / Journal of Mathematical Economics 68 (2017) 55–61

where X (t ) = Rl+ for every t ∈ T ; w(t ) = wi for all t ∈ Ii ; the preference relation ≻t is represented by Ut which satisfies that for any t ∈ Ii and x, y ∈ X (t ), Ut (x) > Ut (y) if and only if ui (x) > ui (y); for each coalition S ∈ F (where F is the set of ′ all measurable subsets of T ), define Y S = Y S , where S ′ = {i ∈ T N N : µ(S ∩ Ii ) > 0} (thus, Y = Y = Y ) with µ being Lebesgue measure and β(t , p) = 1 for any t ∈ T and any p ∈ P. An allocation (or ‘‘trade’’) for a continuum economy is an assignment x for which



[x(t ) − w(t )]dt = y ∈ Y .

(3.1)

T

See Debreu and Scarf (1963) for the corresponding concept in finite production economies. For each allocation x = (xi )i∈N ∈ F (E ), define the step function fx (t ) by fx (t ) = xi

Then  fx (i t ) is ian allocation in EC . In fact, x ∈ F (E ) implies that i∈N (x − w ) ∈ Y . By assumption (P.1),



[fx (t ) − w(t )]dt = T

The following fact proved in Liu (submitted for publication) shows that every competitive allocation belongs to the core in a continuum economy. Theorem 3.3 (Liu, 2015). Any competitive allocation belongs to the core in a coalition production economy with a continuum of agents. To prove the next theorem, we need the following lemma which is the lemma together with its remark by García-Cutrín and Hervés-Beloso (1993). Lemma 3.4 (García-Cutrín and Hervés-Beloso, 1993). Let ≽ be convex and continuous preference relation. If S ⊆ T has positive measure, g : S → Rl+ is an integrable function and x ∈ Rl+ is such that g (t ) ≻ x (or g (t ) ≽ x) for all t ∈ S, then 1



µ(S )

if t ∈ Ii .

1 i∈N

n

(xi − wi ) =

1 n i∈N

(xi − wi ) ∈ Y

which implies that fx (t ) is an allocation in EC by (3.1). Remark B. The assumptions (A.1)–(A.3) and (P.1) for an economy E clearly imply the assumptions (A.1)–(A.4) and (P.1) given in Liu and Zhang (2016) for the associated continuum economy EC .

57

g (t )dt ≻ x

 or

S





1

µ(S )

g (t )dt ≽ x resp. . S

In what follows, each allocation x(t )  in EC yields an allocation x′ = (x1 , x2 , . . . , xn ) in E with xi = n I x(t )dt for 1 ≤ i ≤ n. i The proof of the following theorem is motivated by the proof of Theorem 1 from García-Cutrín and Hervés-Beloso (1993). Theorem 3.5. Let E be an economy satisfying assumptions (A.3) and (P.1). If (p, x, y) is a competitive equilibrium for EC , then (p, x′ , ny) is a competitive equilibrium for E . Proof. Let (p, x, y) be a competitive equilibrium for EC . Then

An allocation y dominates an allocation x via a coalition S ∈ F if y(t ) ≻t x(t ) for each t ∈ S and





and p · y is maximum on Y. By Remark C, p · y = 0. For x′ = (x1 , x2 , . . . , xn ) with xi = n Ii x(t )dt = |I1i | Ii x(t )dt for 1 ≤ i ≤ n,  since Y is a cone by assumption (P.1) and T [x(t ) − w(t )]dt = y ∈

[y(t ) − w(t )]dt ∈ Y S . S

Definition 3.1. The core of a continuum economy is the set of all allocations that are not dominated via any nonnull coalition. ′

′

For convenience, denote sup{p · y : y ∈ Y } by sup p · Y . The following concept of competitive equilibrium for a continuum economy is defined in Liu and Zhang (2016) (also see Hildenbrand, 1968 or Hildenbrand, 1970).

[x(t ) − w(t )]dt = y ∈ Y T

Y , we have

  1  i i i i (x − w ) = n (x − w ) i∈N

i∈N



n

[x(t ) − w(t )]dt = ny ∈ Y .

=n T

Definition 3.2. A competitive equilibrium (Walrasian equilibrium) of a continuum economy consists of a price vector p ∈ Rl+ with p ̸= 0, an allocation x(t ), and a production y ∈ Y such that (i) T [x(t ) − w(t )]dt = y ∈ Y S (ii) p · y = sup  p · Y and for any coalition S ∈ F , sup p · Y ≤ (sup p · Y ) S β(t , p)dt; (iii) for almost every (a.e.) trader t, x(t ) is maximal with respect to ≻t in t’s budget set Bp (t ) = {x ∈ X (t ) : p · x ≤ p · w(t ) + (sup p · Y )β(t , p)}, that is, for almost every t ∈ T ,



p · x(t ) ≤ p · w(t ) + (sup p · Y )β(t , p) and

v ≻t x(t ) implies p · v > p · w(t ) + (sup p · Y )β(t , p).

(3.2)

Remark C. Similar to Remark A, for a continuum economy under the assumption (P.1), we have that for any price vector p at equilibrium and for any coalition S ∈ F , p · y ≤ 0 for any y ∈ Y S . It follows that the maximum profit sup{p · y : y ∈ Y S } at price p on Y S is zero for each S ∈ F . Thus, together with assumption (A.1), the budget set in Definition 3.2 for competitive equilibrium becomes Bp (t ) = {x ∈ Rl+ : p · x ≤ p · w(t )} and (3.2) is reduced to p · x(t ) ≤ p · w(t ) and

v ≻t x(t ) implies p · v > p · w(t ). (3.3)

Thus, x′ is an allocation for E . Since p · y = 0, p · ny = 0. Thus, by Remark A, the profit is maximized on Y at ny for the economy E . Next, we show that each xi is in the budget set Bp (i) = {x ∈ Rl+ : p · x ≤ p · wi }. Since x(t ) is in the budget set Bp (t ) = {x ∈ Rl+ : p · x ≤ p · w(t )} (see Remark C) for all t ∈ Ii , we have for each 1 ≤ i ≤ n, p · x(t ) ≤ p · w(t ) and p · xi = p · n



x(t )dt ≤ p · n Ii



w(t )dt = p · wi .

Ii

To show that (p, x′ , ny) is a competitive equilibrium for E , it suffices to show that for 1 ≤ i ≤ n, ui (vi ) > ui (xi ) implies p · vi > p · wi . Let ui (vi ) > ui (xi ) and define v(t ) by setting v(t ) = vi for all t ∈ Ii . We claim that there exists t ′ ∈ Ii such that Ut ′ (v(t ′ )) > Ut ′ (x(t ′ )), where Ut is the continuous and strong convex function representing the preference relation ≻t in EC . Suppose, for otherwise, that Ut (vi ) = Ut (v(t )) ≤ Ut (x(t )) for all t ∈ Ii . By assumption (A.3) (strong concavity implies concavity) and Lemma 3.4, we have Ut (vi ) ≤ Ut



1

| Ii |



x(t )dt



= Ut (xi )

Ii

which implies ui (vi ) ≤ ui (xi ) by the construction, contradicting ui (vi ) > ui (xi ). Thus the claim holds. Since (p, x, y) is a competitive

58

J. Liu / Journal of Mathematical Economics 68 (2017) 55–61

equilibrium for EC , Ut ′ (v(t ′ )) > Ut ′ (x(t ′ )) implies that p · v(t ′ ) > p · w(t ′ ). It follows that p · vi > p · wi as v(t ′ ) = vi and w(t ′ ) = wi . Thus, (p, x′ , ny) is a competitive equilibrium for E .  The following concepts of objections, counterobjections, and bargaining sets introduced in Liu and Zhang (2016) are continuous versions of the corresponding concepts for finite economies. Definition 3.6. An objection to an allocation x is a pair (S , y), where S ∈ F and y : S → Rl+ , such that (a) S [y(t ) − w(t )]dt ∈ Y S , (b) y(t ) 0.



Definition 3.7. Let (S , y) be an objection to an allocation x. A counterobjection to (S , y) is a pair (Q , z), where Q ∈ F and z : Q → Rl+ , such that (a) Q [z(t ) − w(t )]dt ∈ Y Q , (b) µ(Q ) > 0, (c) z(t ) ≻t y(t ) for a.e. t ∈ S ∩ Q and z(t ) ≻t x(t ) for a.e. t ∈ Q \ S.



Definition 3.8. An objection (S , y) is said to be justified if there is no counterobjection to it. The bargaining set of a continuum economy is the set of all allocations which have no justified objection. Clearly, the core of an economy E is contained in its bargaining set. In 1989, Mas-Colell (1989) proved that the bargaining set coincides with the set of competitive allocations in an exchange economy with a continuum of traders satisfying some standard assumptions. This result is extended to continuum coalition production economies by Liu and Zhang (2016) recently. 4. Coincidence of the Aubin bargaining set and the set of competitive allocations Recall that the bargaining set coincides with the set of competitive allocations in a continuum economy satisfying assumptions (A.1)–(A.4) and (P.1). It is well-known that in a finite exchange economy, this coincidence no longer holds. Recently, Hervés-Estévez and Moreno-García (2016a) introduced the concept of Aubin bargaining set for exchange economies. Motivated by ideas from García-Cutrín and Hervés-Beloso (1993) and Hervés-Estévez and Moreno-García (2016a), we will show that the Aubin bargaining set coincides with the set of competitive allocations in an economy under some standard assumptions in this section, which extends the same conclusion for finite exchange economies in Hervés-Estévez and Moreno-García (2016a). First, we introduce the concept of Aubin bargaining set for an economy which extends the corresponding concept for exchange economies given by Hervés-Estévez and Moreno-García (2016a), motivated by Aubin’s veto mechanism (Aubin, 1981) and the concept of fuzzy bargaining sets from Liu and Liu (2012). Definition 4.1. An Aubin objection to an allocation x is a pair (S , y), where S ∈ N and y is defined on S, for which there exist 0 < αi ≤ 1 for each i ∈ S, such that i i S (a) i∈S αi (y − w ) ∈ Y , i i i i (b) u (y ) ≥ u (x ) for each i ∈ S with at least one of these inequalities being strict.



Definition 4.2. Let (S , y) be an Aubin objection to an allocation x. An Aubin counterobjection to (S , y) is a pair (Q , z), where Q ∈ N and z is defined on Q , for which there exist 0 < λi ≤ 1 for each i ∈ Q , such that (a)



i∈Q

λi (zi − wi ) ∈ Y Q ,

(b) ui (zi ) > ui (yi ) for each i ∈ S ∩ Q and ui (zi ) > ui (xi ) for each i ∈ Q \ S. Definition 4.3. An Aubin objection (S , y) is said to be justified if there is no Aubin counterobjection to it. The Aubin bargaining set BA (E ) of an economy E is the set of all allocations which have no justified Aubin objection. Clearly, the Aubin bargaining set BA (E ) is a subset of the bargaining set B (E ) in an economy E . Recall from Aubin (1981) that a fuzzy coalition is a vector s = (s1 , s2 , . . . , sn ) with 0 ≤ si ≤ 1 for each 1 ≤ i ≤ n. Florenzano (1990) defined the fuzzy core CF (E ) of an economy E to be the set of all allocations which cannot be blocked by any fuzzy coalition, where an allocation x  is blocked by i a fuzzy coalition s means that there exists y ∈ X = i∈N X such  i i S that i∈N si (y − w ) ∈ Y with S = car (s) = {i ∈ N : si > 0}, and ui (yi ) ≥ ui (xi ) for each i ∈ S with at least one of the inequalities being strict (i.e., an Aubin objection). Thus, it is clear that the fuzzy core CF (E ) is a subset of the Aubin bargaining set BA (E ) for an economy E . We make a brief remark here on (fuzzy) bargaining sets and (fuzzy) cores. Bargaining sets and cores, as important solutions in game theory and economics, have been studied extensively. The concept of bargaining sets was first introduced by Aumann and Maschler (1964), see Anderson (1998), Anderson et al. (1997), Aubin (1979), Aumann and Maschler (1964), Hervés-Estévez and Moreno-García (2016a,b), Mas-Colell (1989), and Shapley and Shubik (1984) for selected literature on bargaining sets. The fuzzy core of an economy can be viewed as a refinement to the core of the economy by allowing agents to cooperate at a different participation level (from 0% to 100%), thereby with more blocking power, the Aubin bargaining set does the same to the bargaining set for an economy. The following fact is standard. Lemma 4.4. For an economy E satisfying assumptions (A.2), (A.3), and (P.1), any competitive allocation of E is in the fuzzy core of E . Proof. Suppose (x, p) is a competitive equilibrium but x is not in the fuzzy core CF (E ). Then there exists a fuzzy coalition s = (s1 , s2 , . . . , sn ) such by s, that is, there exists  that x i is blocked i S y ∈ X such that i∈N si (y − w ) ∈ Y , where S = car (s) = {i ∈ N : si > 0}, and ui (yi ) ≥ ui (xi ) for each i ∈ S with at least  one of the inequalities being strict. By Remark A, we have p · i∈N si (yi − wi ) ≤ 0. On the other hand, since x is competitive, p · xi = p · wi for all i ∈ N and ui (yi ) > ui (xi ) implies p · yi > p · xi . By assumptions (A.2) ui (yi ) ≥ ui (xi ) implies p · yi ≥ p · xi .  and (A.3), i Thus, we have p · i∈N si (y − wi ) > 0, a contradiction.  Now, we are ready to prove our first main result. Our approach is similar to that used in Hervés-Estévez and Moreno-García (2016a), with the following main differences on how to overcome the difficulties caused by productions involved: First, we have to use Theorem 3.5 from Liu and Zhang (2016)—the extension of the corresponding result by Mas-Colell (1989) which is used in HervésEstévez and Moreno-García (2016a); Second, we need to pay special attention to the fact that an allocation in a continuum economy E here must satisfy (3.1) and (Aubin) objections or (Aubin) counterobjections must satisfy condition (a) in the corresponding definitions, which are not required in exchange economies considered by Hervés-Estévez and Moreno-García (2016a). For example, extra effort is needed for the requirement in (4.5). Theorem 4.5. Let E be an economy satisfying assumptions (A.1) – (A.3) and (P.1). Then the set of competitive allocations and the Aubin bargaining set coincide in E .

J. Liu / Journal of Mathematical Economics 68 (2017) 55–61

Proof. By Lemma 4.4 and the fact that the fuzzy core CF (E ) is a subset of the Aubin bargaining set BA (E ), we conclude that the set of competitive allocations of E is a subset of the Aubin bargaining set BA (E ). We now show that the Aubin bargaining set BA (E ) is a subset of the set of competitive allocations of E . Suppose x ∈ BA (E ). We will show that x is a competitive allocation. Let fx be the step function obtained from x. We first show that fx is in the bargaining set of EC . Assume that (S , g ) is an objection to fx in EC . Then there exists y ∈ Y S such that



[g (t ) − w(t )]dt = y, S

Ut (g (t )) ≥ Ut (fx (t ))

for a.e. t ∈ S and µ{t ∈ S : Ut (g (t )) > Ut (fx (t ))} > 0.

(4.1)

Let Si = S ∩ Ii for 1 ≤ i ≤ n and S ′ = {i ∈ N |µ(Si ) > 0}. By (4.1), there exist k ∈ N and A ⊆ Sk such that µ(A) > 0 and Ut (g (t )) > Ut (fx (t )) for every t ∈ A. Let a be the allocation given by ai = µ(1S ) S g (t )dt for each i ∈ S ′ . Then i



i



Recall that for each i ∈ S ′ , fx (t ) = xi and Ut (g (t )) ≥ Ut (fx (t )) for all t ∈ Si . Similar to the argument in the proof of Theorem 3.5, by  assumption (A.3) and Lemma 3.4, Ut (ai ) = Ut ( µ(1S ) S g (t )dt ) ≥ i

i

Ut (xi ) for each i ∈ S ′ and Ut (ak ) = Ut ( µ(1S ) S g (t )dt ) > Ut (xk ). It k k follows that ui (ai ) ≥ ui (xi ) for each i ∈ S ′ and uk (ak ) > uk (xk ). Thus, (S ′ , a) is an Aubin objection to x in the finite economy E . Since x is in the Aubin bargaining set BA (E ), there exists an Aubin counterobjection (Q , z) to the objection (S ′ , a), that is, there exist 0 < λi ≤ 1 for each i ∈ Q , such that



λi (zi − wi ) ∈ Y Q ,

(4.2)

i∈Q

ui (zi ) > ui (ai )

for each i ∈ S ′ ∩ Q and

ui (zi ) > ui (xi ) for each i ∈ Q \ S ′ .

(4.3)

By assumption (A.3), Lemma 3.4, and (4.3), for every i ∈ S ′ ∩ Q , there exists Ai ⊆ Si such that µ(Ai ) > 0 and ui (zi ) > Ut (g (t ))

for every t ∈ Ai .

(4.4)

Let b = min{µ(Ai )|i ∈ S ∩ Q } and take M > 0 large enough so that λi βi = M ≤ min{b, 1n } for every i ∈ Q . Let Ti ⊆ Ai with µ(Ti ) = βi for each i ∈ S ′ ∩ Q , and Ti ⊆ Ii with µ(Ti ) = βi for each i ∈ Q \ S ′ . Now, set T ′ = ∪i∈Q Ti and define the function h(t ) on T ′ by setting h(t ) = zi for all t ∈ Ti . Since Y Q is a cone by assumption (P.1), it follows from (4.2) that ′

 T′

Since the fuzzy core is a subset of the Aubin bargaining set in an economy, Lemma 4.4 and Theorem 4.5 imply the next fact immediately. Theorem 4.6. Let E be an economy satisfying assumptions (A.1) – (A.3) and (P.1). Then the set of competitive allocations and the fuzzy core coincide in E . We remark here that Theorems 3.5, 4.5 and 4.6, together with Corollary 3.4 in Liu (submitted for publication), will imply the existence for fuzzy cores and Aubin bargaining sets. 5. A limit theorem on bargaining sets

′

[g (t ) − w(t )]dt = y ∈ Y S = Y S . S

i∈S ′



Thus, (T ′ , h) forms a counterobjection to the objection (S , g ) for fx in the continuum economy EC . It follows that fx is in the bargaining set of EC . By Theorem 3.5 in Liu and Zhang (2016) and Remark B, fx is a competitive allocation for EC . It follows from Theorem 3.5 here that x is a competitive allocation for E . Therefore, we have shown that the Aubin bargaining set BA (E ) is a subset of the set of competitive allocations of E . This completes the proof of the theorem. 

i

µ(Si )(a − w ) = i

59

[h(t ) − w(t )]dt =



βi (z − w ) = i

i

i∈Q

=

1  M i∈Q

 λi i∈Q

M

λi (zi − wi ) ∈ Y Q .

(z − w ) i

i

(4.5)

for every t ∈ Ti with i ∈ S ′ ∩ Q ,

Ut (h(t )) = ui (zi ) > ui (xi ) = Ut (fx (t )) for every t ∈ Ti with i ∈ Q \ S ′ , that is, Ut (h(t )) > Ut (g (t )) for every t ∈ T ′ ∩ S , Ut (h(t )) > Ut (fx (t ))

Definition 5.1. The objection (S , y) to the allocation x is competitive if there is a price system p ̸= 0 such that for a.e. t ∈ T : (i) p · v ≥ p · w(t ) for v ∈ Rl+ satisfying v
By (4.3) and (4.4), we have Ut (h(t )) = ui (zi ) > Ut (g (t ))

A classical theorem by Debreu and Scarf (1963) shows that when the set of economic agents is replicated, the set of core allocations of the replica economy converges to the set of competitive allocations. However, Anderson et al. (1997) provided an example of an economy in which the Mas-Colell bargaining set does not shrink to the set of competitive allocations when the economy is replicated. In this section, we will provide a convergence theorem on Mas-Colell bargaining sets for finite coalition production economies under certain standard assumptions which extends the corresponding result for exchange economies by Hervés-Estévez and Moreno-García (2016b). First, we recall the following concept of competitive objection for a continuum economy (see Liu and Zhang, 2016 and Mas-Colell, 1989).

for every t ∈ T \ S . ′

The first index of consumer (i, q) refers to the type of the individual and the second index distinguishes different individuals of the same type. It is assumed that all consumers of type i are identical in terms of their consumption sets, endowments, and utility functions. Let S be a nonempty subset of Nr . An allocation (x(i,q) )(i,q)∈S is S-attainable in the economy Er if

 (i,q)∈S

(x(i,q) − w(i,q) ) ∈ Y S , ′

(5.1)

60

J. Liu / Journal of Mathematical Economics 68 (2017) 55–61

where S ′ = {i ∈ N |(i, q) ∈ S }, x(i,q) ∈ X i and w(i,q) = wi for every q. Thus, (5.1) can be written as



(i,q)

x

−



i∈S ′ q∈S (i)

S′

|S (i)|w ∈ Y , i

(5.2)

i∈S ′

where S (i) = {q ∈ {1, 2, . . . , r }|(i, q) ∈ S } and |S (i)| denotes the number of elements in S (i). We need the following additional assumption on production sets. (P.2) For each S ∈ N , Y S is comprehensive, that is, given y ∈ Y S , if y′ ≤ y, then y′ ∈ Y S . Lemma 5.2 implies the next lemma for finite coalition production economies, where rx denotes the r-fold repetition of an allocation x from a finite economy E , that is, rx = (x(i,q) )(i,q)∈Nr , where x(i,q) = xi for each 1 ≤ q ≤ r and 1 ≤ i ≤ n. Lemma 5.3. Let E be an economy satisfying assumptions (A.1) – (A.3), (P.1) and (P.2). If x is an allocation which is not competitive in E , then there is a competitive objection (S , y) to rx in the r-fold replica Er of E for some r ≥ 1.

Since rki , i ∈ S ′ , are nonnegative rational numbers, we can express rki =

mik

for each i ∈ S ′ , where mik and nik are nonnegative integers.

nik

By multiplying both sides of Eq. (5.7) with we obtain





i∈S ′

nik and using (P.1),

′

pik [yi∗ − wi ] ∈ Y S ,

(5.8)

i∈S ′

where pik are nonnegative integers. Let r = max{pik : i ∈ S ′ } and (i,q)

define y = (y

)(i,q)∈S ∗ on S ∗ = {(i, q) : i ∈ S ′ , q ≤ pik } by setting y = y∗ for each 1 ≤ q ≤ pik and i ∈ S ′ . By Lemma 3.4 and (5.4)– (5.6), (5.8), it is easy to verify that (S ∗ , y) is a competitive objection (i,q)

i

to rx in the r-fold replica Er of E .



The next lemma follows from an easy standard argument. Lemma 5.4. Let E be an economy satisfying assumptions (A.1) –(A.3) and (P.1). Then every competitive objection (S , y) to an allocation x = (x(i,q) )(i,q)∈Nr in the r-fold replica Er is justified.

Proof. Let x be an allocation in E which is not competitive. By Theorem 3.5, fx (t ) is not competitive in the corresponding continuum economy EC (see Section 3), where fx (t ) is defined by

Proof. Let E be an economy satisfying assumptions (A.1)–(A.3) and (P.1). Assume that (S , y) is a competitive objection to an allocation x = (x(i,q) )(i,q)∈Nr in the r-fold replica Er at price p, where S = {(i, q) : i ∈ N , 0 ≤ q ≤ ri ≤ r }. Suppose, to the contrary, that there exists a counterobjection (Q , z) to (S , y), that is, there exists z defined on Q = {(i, q) : i ∈ N , 0 ≤ q ≤ qi } ⊆ Nr such that

fx (t ) = xi

(a)

if t ∈ Ii .

By Remark B, EC satisfies the assumptions (A.1)–(A.3) and (P.1). It follows from Lemma 5.2 that there is a competitive objection (S , y) to fx (t ) in EC , that is, S ∈ F and y : S → Rl+ , such that



[y(t ) − w(t )]dt ∈ Y S ,

(5.3)



(i,q)∈Q (z

(i,q)

and there is a price system p ̸= 0 such that for a.e. t ∈ T ,

p· (5.5)

p · v ≥ p · w(t )

for v ∈ Rl+ satisfying v
(5.6)

Set Si = S ∩ Ii for each 1 ≤ i ≤ n and let S = {i : µ(Si ) > 0}. By (5.4), S ′ ̸= ∅. Define y∗ on S ′ by ′

yi∗ =



1

µ(Si )

y(t )dt

for i ∈ S ′ .

Si

Recall that w(t ) = wi for t ∈ Si ⊆ Ii . It follows from (5.3) that



µ(Si )[yi∗ − wi ] =

 Si

i∈S ′

i∈S ′

y(t )dt −





w(t )dt



S

For each i ∈ S , since 0 < µ(Si ) ≤ 1, there exists a sequence { } of rational numbers between 0 and 1 such that

rki k≥1

(1) rki → µ(Si ) as k → ∞; (2) If yi∗ − wi ≤ 0, then rki ≥ µ(Si ) and rki > rki +1 for all k ≥ 1; If yi∗ − wi > 0, then rki ≤ µ(Si ) and rki < rki +1 for all k ≥ 1. It follows that for each k ≥ 1,



rki [yi∗ − wi ] ≤



′

µ(Si )[yi∗ − wi ] = z ∈ Y S .

i∈S ′

i∈S ′

By (P.2), we have z′k =

 i∈S ′

′

rki [yi∗ − wi ] ∈ Y S .

for each (i, q) ∈ Q \ S .

(z(i,q) − w(i,q) ) > 0.

(i,q)∈Q

On the other hand, by Remark A, p·



(z(i,q) − w(i,q) ) = p · g ≤ 0,

(i,q)∈Q

a contradiction. Thus, the lemma holds.



We say that an allocation x of a finite economy E is in the bargaining set of the r-fold replica Er of E if rx ∈ B (Er ). The following theorem shows that the (Mas-Colell) bargaining set shrinks to the set of competitive allocations in an economy E under some standard conditions when E is replicated.

competitive allocations of E .

′

′

z′k =



for each (i, q) ∈ S ∩ Q ;

Theorem 5.5. Let E be an economy satisfying assumptions (A.1) – (A.3), (P.1) and (P.2). Then ∩r ≥1 B (Er ) coincides with the set of

Si



y(t ) − w(t ) dt = z ∈ Y S = Y S .

=

(i,q)

>p·w =p·w i

It follows that

for v ∈ Rl+ satisfying v
′

p · z(i,q) > p · wi = p · w(i,q)

y(t ) 0; (5.4)

p · v ≥ p · w(t )

− w(i,q) ) = g ∈ Y Q , where Q ′ = {i ∈ N : qi >

0}, (b) ui (z(i,q) ) > ui (y(i,q) ) for each (i, q) ∈ S ∩ Q and ui (z(i,q) ) > ui (x(i,q) ) for each (i, q) ∈ Q \ S. Since (S , y) is a competitive objection to x at price p, we have p·z

S

(i,q)

(5.7)

Proof. Let E be an economy satisfying assumptions (A.1)–(A.3), (P.1) and (P.2). By Lemma 4.4, every competitive allocation is in the fuzzy core of E which is a subset of the core of E . As the core of E is contained in the bargaining set B (Er ) for every r ≥ 1, we have that the set of competitive allocations of E is contained in ∩r ≥1 B (Er ). On the other hand, suppose that x ∈ ∩r ≥1 B (Er ), but x is not competitive in E . Then, by Lemma 5.3, there is a competitive objection (S , y) to rx in the r-fold replica Er of E for some r ≥ 1. By Lemma 5.4, (S , y) is a justified objection to rx in Er which implies that x ̸∈ B (Er ), a contradiction. Thus, the theorem follows.  Note that the bargaining set for an economy E defined by Hervés-Estévez and Moreno-García (2016b) is a subset of the standard (Mas-Colell) bargaining set B (E ) for an economy E used here (and in Mas-Colell, 1989). Theorem 5.5 implies Theorem 4.1 in Hervés-Estévez and Moreno-García (2016b).

J. Liu / Journal of Mathematical Economics 68 (2017) 55–61

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