Int J Game Theory DOI 10.1007/s00182-015-0511-9 ORIGINAL PAPER

Coincidence of the Mas-Colell bargaining set and the set of competitive equilibria in a continuum coalition production economy Jiuqiang Liu1,2 · Huihui Zhang3

Accepted: 7 October 2015 © Springer-Verlag Berlin Heidelberg 2015

Abstract Mas-Colell (J Math Econ 18:129–139, 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. We extend this result to continuum coalition production economies and prove that the bargaining set and the set of competitive allocations coincide in a coalition production economy with a continuum of traders under some standard assumptions. As a consequence, we obtain a coincidence theorem for the core and the set of competitive allocations in a coalition production economy which extends the well-known coincidence theorem by Aumann (Econometrica 32:39–50, 1964). Keywords Bargaining sets · Cores · Competitive equilibrium · Coalition production economy · Exchange economy

1 Introduction Cores and equilibria are important solutions for economies. Arrow and Debreu (1954) established the celebrated existence theorem of competitive equilibria in

B

Jiuqiang Liu [email protected] Huihui Zhang [email protected]

1

School of Management Engineering, Xi’an University of Finance and Economics, Xi’an 710100, Shaanxi, People’s Republic of China

2

Department of Mathematics, Eastern Michigan University, Ypsilanti, MI 48197, USA

3

Faculty of Mathematics and Statistics, Central China Normal University, Wuhan 430079, People’s Republic of China

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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, see Aumann (1966), Bewley (1972), Greenberg et al. (1979), Hildenbrand (1970), Podczeck and Yannelis (2008), and Schmeidler (1969) for some of the existence results. 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. Aumann’s coincidence theorem has been extended in several different ways. Armstrong and Richter (1985) studied the equivalence of cores and competitive equilibrium sets for exchange economies in the very general framework of Boolean rings and algebras. Podczeck (2004, 2005) extended Aumann’s theorem to Banach spaces and Banach lattices, but remained in exchange economies. Hildenbrand (1968) extended Aumann’s result to coalition production economies under the condition that production possibility sets are additive, and in 1993, Basile (1993) made an extension to coalition production economies when production possibility sets are finitely additive. 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. Here, we will extend Mas-Colell’s coincidence theorem to continuum coalition production economies and prove that the bargaining set and the set of competitive allocations coincide in a coalition production economy with a continuum of traders under some standard assumptions. As a consequence, we obtain a coincidence theorem for the core and the set of competitive allocations in a coalition production economy which extends the well-known coincidence theorem by Aumann (1964).

2 Preliminaries 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. First, we recall some basic concepts introduced in Aumann (1964) and Liu (2015). A commodity bundle x is a point in the nonnegative orthant Rl+ of Rl , where l is the number of different commodities. The set of traders is the closed interval [0, 1], denoted by T . Let X : T → Rl+

(2.1)

be a measurable consumption correspondence, where X (t) is interpreted as the consumption set of agent t ∈ T . An assignment (of commodity bundles to traders) is a function x ∈ X (i.e., x(t) ∈ X (t) for every t ∈ T ), each coordinate of which is Lebesgue integrable over T . There is an initial assignment (called initial endowment)  w(t) satisfying T w(t) dt > 0, where integrals are taken through Lebesgue measure

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Coincidence of the Mas-Colell bargaining set and the set…

  μ, so integral S f dt is the same as S f dμ throughout the paper. For each trader t, there is a preference relation t on Rl+ . We need to make the following standard assumptions on X (t), w(t), and the preference relation t . (A.1) For every t ∈ T , X (t) = Rl+ and w(t) > 0. (A.2) Desirability (of the commodities): x ≥ y implies x t y, where t satisfies the transitivity and irreflexivity. (A.3) Continuity (of the commodities): For each y ∈ X (t), the sets {x : x t y} and {x : y t x} are open (relative to X (t)) for a.e. t ∈ T . (A.4) Measurability: If x and y are assignments, then the set {t : x(t) t y(t)} is Lebesgue measurable in T . A coalition of traders is a Lebesgue measurable subset S of T ; if it is of measure 0, it is called null. Let F be the set of all Lebesgue measurable subsets (coalitions) of  T . Denote the normalized price set P = { p ∈ Rl+ : li=1 pi = 1}. A coalition production economy with a continuum of traders is E = (Rl , X, (t , w(t))t∈T , (Y S ) S∈F , β(t, p)), where X : T → Rl+ is a measurable consumption correspondence, each Y S ⊆ Rl is the production set of the firm (coalition) S ∈ F with Y T = Y being the total production set (where inputs into production appear as negative components of y ∈ Y S and outputs as positive components), and β(t, p) : T × P →  [0, 1] is the profit distribution function for the total production set Y such that T β(t, p) dt = 1 and β(t, p) is continuous with respect to p, where each agent t receives profit share ( p · y)β(t, p) from the total profit p · y at production y ∈ Y and price vector p ( p · y is the inner product of vectors p and y). The profit distribution function β(t, p) plays a similar rule as the profit share coefficients αi j in Arrow–Debreu’s finite economic model in Arrow and Debreu (1954). As pointed out in Liu (2015), this model includes the economic models studied by Greenberg et al. (1979) and Hildenbrand (1970): the model is more general than the model studied by Hildenbrand  (1970) in which it is assumed that the production sets are exists a measurable produccountably additive (i.e., i∈I Y Si = Y ∪i∈I Si ) so that there  tion correspondence Y defined on T satisfying Y S = S Y dt for every coalition S ∈ F (one can define the profit distribution function β(t, p) by β(t, p) = max{1p·Y } π( p, ·) if max{ p · Y } = 0 and 1 otherwise, with π( p, ·) being the profit function there); our model also includes the production economy model considered by Greenberg et al. (1979) which has a measurable production correspondence Y defined on T (where  production sets Y S on coalitions S ∈ F can be defined by Y S = S Y dt which are also additive). In general, as pointed out by Boehm (1974), for any two coalitions S1 and S2 , the outcome of two separate decisions y1 ∈ Y S1 and y2 ∈ Y S2 will not be related in any specific way to a production decision by the coalition S = S1 ∪ S2 . Our model is in a general setting where the production technology is given by a collection of nonempty subsets ({Y S }, Y ) of the commodity space Rl , there may not exist a measurable production correspondence.

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An exchange economy is a coalition production economy with Y S = {0} for every coalition S ∈ F. An allocation (or “trade”) is an assignment x for which  [x(t) − w(t)] dt = y ∈ Y.

(2.2)

T

See Debreu and Scarf (1963) for the corresponding concept in finite production economies. An allocation y dominates an allocation x via a coalition S if y(t) t x(t) for each t ∈ S and  [y(t) − w(t)] dt ∈ Y S . S

Definition 2.1 The core of an economy E with a continuum of traders is the set of all allocations that are not dominated via any nonnull coalition. Throughout the paper, we denote max{ p · y : y ∈ Y } by max p · Y . The following concept of competitive equilibrium is a more general version of the one defined in Hildenbrand (1968) or Hildenbrand (1970): the corresponding concept there follows by setting β(t, p) = max{1p·Y } π( p, ·) if max{ p · Y } = 0 and 1 otherwise with π( p, ·) being their profit function. Definition 2.2 A competitive equilibrium (Walrasian equilibrium) of a coalition production economy E with a continuum of traders 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 (ii) p ·y = max p · Y and for any coalition S ∈ F, max p · Y S ≤ (max 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 B p (t) = {x ∈ X (t) : p · x ≤ p · w(t) + (max p · Y )β(t, p)}, that is, for almost every t ∈ T , p · x(t) ≤ p · w(t) + (max p · Y )β(t, p) and v t x(t) implies p · v > p · w(t) + (max p · Y )β(t, p).

(2.3)

The next fact proved in Liu (2015) by an easy standard method shows that every competitive allocation belongs to the core in a coalition production economy with a continuum of agents. Theorem 2.3 Liu (2015) Any competitive allocation belongs to the core in a coalition production economy with a continuum of agents.

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3 Coincidence theorem of the bargaining set and the set of competitive equilibria Throughout this section, we assume that the preference t is derived from a quasiorder preference-or-indifference relation t , which is assumed to be a quasi-order, i.e., a reflexive, transitive and complete binary relation on Rl+ . From t we derive relations t and ∼t called preference and indifference, respectively, as follows: x t y if x t y but not y t x, x ∼t y if x t y and y t x. For more on the relation between t and t , we refer readers to Gabszewicz and Merttens (1971) and Shitovitz (1973). Remark A Note from Section 4.6 in Debreu (1959) that together with the assumption that t is a quasi-order, the continuity assumption yields the existence of a continuous utility function u t (x) for t (and so t ) on Rl+ for each fixed trader t ∈ T . The following concepts of objections, counterobjections, and bargaining sets are natural extensions of the corresponding concepts for exchange economies given by Mas-Colell (1989). Definition 3.1 An objection to the allocation x is a pair (S, y), where S ∈ F and y is an allocation such that  (a) S [y(t) − w(t)] dt ∈ Y S , (b) y(t) t x(t) for a.e. t ∈ S and μ{t ∈ S : y(t) t x(t)} > 0. Definition 3.2 Let (S, y) be an objection to the allocation x. A counterobjection to (S, y) is a pair (Q, z), where Q ∈ F and z is an allocation such that  (a) Q [y(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 3.3 An objection (S, y) is said to be justified if there is no counterobjection to it. The bargaining set B(E) of the economy E is the set of all allocations which have no justified objection. Clearly, the core of the economy E is contained in the bargaining set B(E). MasColell (1989) proved that following well-known result. Theorem 3.4 (Mas-Colell 1989) For an exchange economy with a continuum of traders satisfying assumptions (A.1)–(A.4), the bargaining set coincides with the set of competitive allocations. To extend Theorem 3.4 to continuum coalition production economies, we need the following assumption on production sets:

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(P.1) Y S is a convex cone containing the origin for each coalition S ⊆ T and Y ∩Rl+ = {0} (impossibility of free production). Remark B As pointed out by Remark C in Liu (2015) [also remarked by Debreu and Scarf (1963) for finite production economy], for a coalition production economy with a continuum of traders under the assumption (P.1), we have that for any price vector p and for any coalition S ∈ F, p · y ≤ 0 for any y ∈ Y S . For otherwise, if p · y > 0 for some y ∈ Y S , then cy ∈ Y S for any c ≥ 0 as the production set Y S is a convex cone and the profit p · (cy) = c( p · y) approaches to the positive infinity as c approaches to the positive infinity, which is impossible. It follows that the maximum profit max{ 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 2.2 for competitive equilibrium becomes B p (t) = {x(t) ∈ Rl+ : p · x ≤ p · w(t)} and (2.3) is reduced to p · x(t) ≤ p · w(t) and v t x(t)

implies p · v > p · w(t).

(3.1)

The following theorem extends Theorem 3.4 to continuum coalition production economies. Theorem 3.5 For a coalition production economy E with a continuum of traders satisfying assumptions (A.1)–(A.4) and (P.1), the bargaining set coincides with the set of competitive allocations. We will prove Theorem 3.5 along the same line as the proof of Theorem 3.4 by Mas-Colell (1989) through competitive objections defined below. Definition 3.6 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 t y(t), t ∈ S; (ii) p · v ≥ p · w(t) for v ∈ Rl+ satisfying v t x(t), t ∈ T \S. Lemma 3.7 For a coalition production economy with a continuum of traders satisfying assumptions (A.1)–(A.4) and (P.1), every competitive objection (S, y) to an allocation x is justified. Proof Let p = ( p 1 , p 2 , . . . , pl ) = 0 be the price vector associated with the competitive objection (S, y). We first prove the following claim: Claim p > 0 (i.e., pi > 0 for each 1 ≤ i ≤ l). Since (S, y) is a competitive objection, we have p · y(t) ≥ p · w(t) for a.e. t ∈ S

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(3.2)

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which implies p ≥ 0 by the desirability assumption (A.2) and condition (i) for competitive objection. On the other hand, since (S, y) is an objection, there exists y ∈ Y S  such that S [y(t) − w(t)] dt = y . By Remark B, 

[y(t) − w(t)] dt = p · y ≤ 0.

p· S

It follows from (3.2) that p · y(t) = p · w(t) for a.e. t ∈ S.

(3.3)

Suppose, to the contrary, that pi = 0 for some i ≤ l, say p 1 = 0.  By assumption (A.1), w(t) > 0 for all t ∈ S which implies that p S y(t) dt = p · S w(t) dt > 0 as  p ≥ 0. Thus, there exists 2 ≤ i ≤ l, say i = 2, such that p 2 > 0 and S y2 (t) dt > 0 which implies that y2 (t) > 0 for all t ∈ S ⊆ S with μ(S ) > 0 (where the superscript 2 indicates the second coordinate of the corresponding vector). For t ∈ S, set y∗ (t) = y(t) + (1, 0, . . . , 0). Since p 1 = 0, we have p · y∗ (t) = p · y(t) for all t ∈ S. For each t ∈ S , since y∗ (t) ≥ y(t), the desirability assumption (A.2) implies ∗ y (t) t y(t). By the continuity assumption (A.3), there exists 0 < δt < y2 (t) such that y∗ (t) + (0, −δt , 0, . . . , 0) t y(t) and we have from condition (i) for competitive objection that for each t ∈ S , p · [y∗ (t) + (0, −δt , 0, . . . , 0)] ≥ p · w(t). But, by (3.3) and the identity p · y∗ (t) = p · y(t) for all t ∈ S, we have for a.e. t ∈ S , p · [y∗ (t)+(0,−δt , 0, . . . , 0)] = p · y∗ (t)−δt p 2 = p · y(t)−δt p 2 < p · y(t) = p · w(t), a contradiction. Thus, the claim holds. Suppose there is a counterobjection (Q, z) to (S, y). Then there exists y ∈ Y Q such that  [z(t) − w(t)] dt = y . (3.4) Q

Since (S, y) is a competitive objection and p > 0, together with the continuity assumption (A.3), we have that z(t) t y(t) implies p · z(t) > p · w(t) for a.e. t ∈ S ∩ Q and z(t) t x(t) implies p · z(t) > p · w(t) for a.e. t ∈ Q\S.

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It follows that 



p·

z(t) dt > p · Q

w(t) dt. Q

But, by Remark B and (3.4), we have 





p·

w(t) dt = p · y ≤ 0,

z(t) dt − Q

Q

a contradiction.   The proof of the next lemma is motivated by the proof of Lemma 3.2 from Einy et al. (2001) and ideas from Mas-Colell (1989). We also need the following result by Debreu (1959) ((1) in Section 5.6). Proposition 3.8 (Debreu 1959) Let Z be a compact subset of Rl . If ζ is an upper semicontinuous correspondence from P to Z such that, for every p ∈ P, the set ζ ( p) is non-empty convex and satisfies p · ζ ( p)  0, then there is a p ∈ P such that ζ ( p) ∩ (−Rl+ ) = ∅. Lemma 3.9 Let E be a coalition production economy with a continuum of traders satisfying assumptions (A.1)–(A.4) and (P.1). If x is an allocation which is not competitive in E, then there is a competitive objection (S, y) to x. Proof Assume that x is an allocation which is not competitive. We will construct a competitive objection to x. Let P = { p ∈ Rl+ : li=1 pi = 1}. Then P is compact and convex. By Remark A, we can use a continuous utility function u t : X → R to represent the preference relation t (which is derived from a quasi-order relation t ). By Remark B, under assumption (P.1), the budget set for agent t ∈ T at each p ∈ P is B p (t) = {x ∈ Rl+ : p · x ≤ p · w(t)}. For each p ∈ P and t ∈ T , since the budget set B p (t) is compact, the continuous function u t attains maximum on B p (t). For all p ∈ P and all t ∈ T , define D( p, t) = {z(t) ∈ Rl+ : z maximizes u t on B p (t)}. Then it is easy to see that D( p, t) is convex and closed for each p ∈ P and each t ∈ T . For each p ∈ P, define ⎧ ⎨ D( p, t) F( p, t) = D( p, t) ∪ {w(t)} ⎩ {w(t)}

123

if u t (D( p, t)) > u t (x(t)) if u t (D( p, t)) = u t (x(t)) if u t (D( p, t)) < u t (x(t)).

Coincidence of the Mas-Colell bargaining set and the set…

Then F( p, t) is closed as D( p, t) is closed for each p ∈ P and each t ∈ T . Let l 

α =1+

w j (t) dt,

j=1 T

and let K =

⎧ ⎨ ⎩

l

x ∈ Rl+ :

xj ≤ α

j=1

⎫ ⎬ ⎭

and Kˆ =

⎧ ⎨ ⎩

x∈K :

l

xj = α

j=1

⎫ ⎬ ⎭

Define the correspondence ϕ : P × T → 2 K by  ϕ( p, t) =

F( p, t) ∩ K Kˆ ∩ {λd : d ∈ F( p, t), 0 ≤ λ < 1}

if F( p, t) ∩ K = ∅ if F( p, t) ∩ K = ∅.

For each p ∈ P, define 

 ϕ( p, t) dt −

ψ( p) = T

Clearly, if F( p, t) ∩ K = ∅, then

w(t) dt. T

l

> α for any d ∈ F( p, t) which implies that there exists 0 ≤ λ < 1 such that λd ∈ Kˆ . Thus ψ( p) is a nonempty subset of the compact set K for every p ∈ P. By Lemma 4.1 in Appendix, ψ( p) is convex. By Lemma 4.4 in Appendix, ψ is upper semicontinuous on P. We claim that p · v ≤ 0 for any p ∈ P and any v ∈ ψ( p). Let  p ∈ P. For any v ∈ ψ( p), there exists f(t) ∈ ϕ( p, t) such that v = T f(t) dt − T w(t) dt. By the definition of ϕ( p, t), for any t ∈ T , we have either f(t) = λw(t) or f(t) = λz(t) for some z(t) ∈ D( p, t) ⊆ B p (t) and some 0 ≤ λ ≤ 1. Since we have p · z ≤ p · w(t) for a.e. t ∈ T , it follows that    f(t) dt − w(t) dt ≤ 0. p·v = p· j=1 d

T

j

T

Thus, the claim holds. By Proposition 3.8, there exists p ∗ ∈ P and z ∗ ∈ ψ( p ∗ ) such that z ∗  0. It follows that there exists an integrable function f(t) on T such that f(t) ∈ ϕ( p ∗ , t) for all t ∈ T and   ∗ z = f(t) dt − w(t) dt. (3.5) T

T

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Since z ∗  0, f(t) ∈ / Kˆ . Thus f(t) ∈ F( p ∗ , t) for all t ∈ T . Let S = {t ∈ T : f(t) ∈ D( p ∗ , t) and u t (D( p ∗ , t)) ≥ u t (x(t))} and C( p ∗ ) = {t ∈ T : u t (D( p ∗ , t)) > u t (x(t))}. Then C( p ∗ ) ⊆ S. By the measurability assumption (A.4), both S and C( p ∗ ) are Lebesgue μ-measurable. Since x is not competitive, we have μ(C( p ∗ )) > 0 which implies μ(S) > 0. By the definition of D( p ∗ , t), the desirability assumption (A.2), and the continuity assumption (A.3), we have p ∗ · f(t) ≥ p ∗ · w(t) for all t ∈ S.

(3.6)

By the definition of F( p ∗ , t), we have f(t) = w(t) for t ∈ T \S. It follows from (3.5) that   z ∗ = f(t) dt − w(t) dt. (3.7) S

S

Let y(t) = f(t) −

1 ∗ z for all t ∈ T. μ(S)

We now show that (S, y) is a competitive objection. Since z ∗  0, we have y(t)  f(t)  ∗ 1 l for all t ∈ T and so y(t) ∈ R+ . Since μ(S) S z dt = z ∗ (the integral is taken through Lebesgue measure μ), it follows from (3.7) that 

 y(t) dt − S

w(t) dt = 0 ∈ Y S ∩ Y S

and so y is an allocation satisfying condition (a) in Definition 3.1. Since y(t)  f(t) for all t ∈ T , it follows from the desirability assumption (A.2) that u t (y(t)) ≥ u t (f(t)) = u t (D( p ∗ , t)) ≥ u t (x(t)) for all t ∈ S. Together with the fact μ(C( p ∗ )) > 0, we have that y satisfies condition (b) in Definition 3.1 and so (S, y) is an objection. Moreover, if t ∈ S and v ∈ Rl+ satisfying u t (v) ≥ u t (y(t)), then we have u t (v) ≥ u t (f(t)) = u t (D( p ∗ , t)) and it follows from either (3.6) or the definition of D( p ∗ , t) that p ∗ · v ≥ p · w(t).

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For any t ∈ T \S, we have u t (D( p ∗ , t)) ≤ u t (x(t)). So, if t ∈ T \S and v ∈ Rl+ satisfying u t (v) ≥ u t (x(t)), then u t (v) ≥ u t (D( p ∗ , t)) and it follows from either (3.6) or the definition of D( p ∗ , t) that p ∗ · v ≥ p · w(t). Therefore, (S, y) is a competitive objection.

 

Proof of Theorem 3.5 By Theorem 2.3 and the fact that the core of an economy E is contained in its bargaining set B(E), we have that the set of competitive allocations is a subset of the bargaining set B(E). On the other hand, by Lemmas 3.7 and 3.9, the bargaining set B(E) is contained in the set of competitive allocations of E. Thus, the set of competitive allocations coincides with the bargaining set B(E) in economy E.   Since for a continuum coalition production economy E, the set of competitive equilibria is contained in its core which is contained in its bargaining set, Theorem 3.5 implies immediately the next result. Theorem 3.10 For a coalition production economy with a continuum of traders satisfying assumptions (A.1)–(A.4) and (P.1), the core coincides with the set of competitive allocations. Recall that an exchange economy is a special coalition production economy with Y S = {0} for every coalition S ⊆ T , Theorem 3.10 implies the following well-known result for exchange economies with a continuum of traders proved by Aumann (1964). Theorem 3.11 (Aumann 1964) For an exchange economy with a continuum of traders satisfying assumptions (A.1)–(A.4), the core coincides with the set of competitive allocations. Acknowledgments The authors would like to thank the referees for their many helpful and inspiring suggestions and comments which resulted significant improvements to the paper. We also would like to thank the associate editor for useful comments.

Appendix We first state the well-known Lyapunov Theorem which is needed for the next lemma. Lyapunov Theorem Let (T, F, μ) be an atomless  finite measure space and f is an integrable function from T into Rl . Then the set { S f dμ : S ∈ F} is a convex set in Rl . Lemma 4.1 The correspondence ψ( p) is convex for each p ∈ P. Proof Fix p ∈ P. To show ψ( p) is convex, we need to show that αv1 + (1 − α)v2 ∈ ψ( p) for any v1 , v2 ∈ ψ( p) and any 0 ≤ α ≤ 1.

(4.1)

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Let v1 , v2 be any two members in ψ( p) and α be any constant satisfying 0 ≤ α ≤ 1. By  ∈ ϕ( p, t) such that v1 =  the definitionof ψ( p), there exist f 1 ( p, t), f 2 ( p, t) 2l f ( p, t) dt − w(t) dt and v = f ( p, t) dt − 1 2 2 T T T T w(t) dt. Let g : F  → R be the atomless measure given by  ( f 1 ( p, t), f 2 ( p, t)) dt

g(S) = S

 for each S ∈ F. By Lyapunov  Theorem, the set W = { S ( f 1 ( p, t), f 2 ( p, t)) dt : S ∈ F} is convex. Since 0 and T ( f 1 ( p, t), f 2 ( p, t)) dt are in W , there exists S ∈ F such that 

 ( f 1 ( p, t), f 2 ( p, t)) dt + (1 − α)0  T ( f 1 ( p, t), f 2 ( p, t)) dt . =α



( f 1 ( p, t), f 2 ( p, t)) dt = α S

T

   which implies S f 1 ( p, t) dt = α T f 1 ( p, t) dt and T \S f 2 ( p, t) dt = (1 −  α) T f 2 ( p, t) dt. Define f

( p, t) =



f 1 ( p, t) f 2 ( p, t)

if t ∈ S if t ∈ / S.

Then f

( p, t) ∈ ϕ( p, t) as f 1 ( p, t), f 2 ( p, t) ∈ ϕ( p, t) and so  T w(t) dt ∈ ψ( p). Moreover, 

f

( p, t) dt =



 T

f

( p, t) dt −



f 2 ( p, t) dt   =α f 1 ( p, t) dt + (1 − α) f 2 ( p, t) dt.

T

f 1 ( p, t) dt +

S

T \S

T

T

It follows that 

  f 1 ( p, t) dt − w(t) dt T T    f 2 ( p, t) dt − w(t) dt +(1 − α) T  T   f 1 ( p, t) dt + (1 − α) f 2 ( p, t) dt − w(t) dt =α T T  T  = f

( p, t) dt − w(t) dt ∈ ψ( p).

αv1 + (1 − α)v2 = α

T

T

Thus, (4.1) holds and ψ( p) is convex for each p ∈ P.

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Coincidence of the Mas-Colell bargaining set and the set…

Let F be a set-valued function from T to Rl+ . F is called Borel measurable if its graph, {(x, t)|x ∈ F(t)}, is a Borel subset of Rl+ × T . Lemma 4.2 For fixed p ∈ P, the correspondence ϕ( p, t) is Borel measurable. Proof Fix p ∈ P. We need to prove that {(x, t)|x ∈ ϕ( p, t)} is aBorel subset of Rl+ × T . Note from the definitions that if F( p, t)) ∩ K = ∅, then li=1 d j > α for any d ∈ F( p, t) which implies that Kˆ ∩ {λd : d ∈ F( p, t), λ ≥ 1} = ∅. It follows from the definition of ϕ( p, t) that {(x, t)|x ∈ ϕ( p, t)} = [(F( p, t) ∩ K ) ∪ ( Kˆ ∩ {λd : d ∈ F( p, t), 0 ≤ λ < 1})] ×[{t : u t (D( p, t)) > u t (x(t))} ∪ {t : u t (D( p, t)) = u t (x(t))} ∪{t : u t (D( p, t)) < u t (x(t))}] = [(F( p, t) ∩ K ) ∪ ( Kˆ ∩ {λd : d ∈ F( p, t), 0 ≤ λ < 1})] × {t : u t (D( p, t)) > u t (x(t))} ∪ [(F( p, t) ∩ K ) ∪ ( Kˆ ∩ {λd : d ∈ F( p, t), 0 ≤ λ < 1})] × {t : u t (D( p, t)) = u t (x(t))} ∪ [(F( p, t) ∩ K ) ∪ ( Kˆ ∩ {λd : d ∈ F( p, t), 0 ≤ λ < 1})] × {t : u t (D( p, t)) < u t (x(t))} = [(F( p, t) ∩ K ) ∪ ( Kˆ ∩ {λd : d ∈ F( p, t), λ ≥ 0})] × {t : u t (D( p, t)) > u t (x(t))} ∪ [(F( p, t) ∩ K ) ∪ ( Kˆ ∩ {λd : d ∈ F( p, t), λ ≥ 0})] × {t : u t (D( p, t)) = u t (x(t))} ∪ [(F( p, t) ∩ K ) ∪ ( Kˆ ∩ {λd : d ∈ F( p, t), λ ≥ 0})] × {t : u t (D( p, t)) < u t (x(t))}. Recall that K and Kˆ are closed and F( p, t) is closed for each p ∈ P and each t ∈ T , the set (F( p, t) ∩ K ) ∪ ( Kˆ ∩ {λd : d ∈ F( p, t), λ ≥ 0}) is closed and thus Borel. By the measurability assumption (A.4), the sets {t : u t (D( p, t)) > u t (x(t))}, {t : u t (D( p, t)) = u t (x(t))}, and {t : u t (D( p, t)) < u t (x(t))} are Borel subsets.   Thus, {(x, t)|x ∈ ϕ( p, t)} is a Borel subset of Rl+ × T . We need the following result from Aumann (1965) for the proof of the next lemma. Proposition 4.3 (Aumann 1965) Let Fx (t) be a set-valued function defined for t ∈ T and x ∈ A, all of whose values are bounded by the same integrable point-valued function, and such that Fx is Borel measurable  for each fixed x ∈ A. If Fx (t) is upper semicontinuous in x for each fixed t, then T Fx (t) dt is upper semicontinuous. Lemma 4.4 The correspondence ψ( p) is upper semicontinuous for each p ∈ P. Proof By Lemma 4.2 and Proposition 4.3, we only need to prove that the correspondence ϕ( p, t) is upper semicontinuous for each fixed t ∈ T . Fix t ∈ T . Let pn → p with all pn , p ∈ P, z n (t) → z(t) with z n (t) ∈ ϕ( pn , t) for all n ≥ 1. We need

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to prove that z(t) ∈ ϕ( p, t). For each n ≥ 1, since z n (t) ∈ ϕ( pn , t), there exists an (t) ∈ F( pn , t) and 0 ≤ λn (t) ≤ 1 such that z n (t) = λn (t)an (t).

(4.2)

By the definition of F( pn , t), we have either an (t) = w(t) or an (t) ∈ D( pn , t) for each n ≥ 1. Since 0 ≤ λn (t) ≤ 1 for all n ≥ 1, {λn (t)}n≥1 has a convergent subsequence, say, λn (t) → λ(t). Clearly, 0 ≤ λ(t) ≤ 1. It follows from (4.2) that an (t) → a(t) ∈ Rl+ and z(t) = λ(t)a(t). If an (t) = w(t) for infinitely many n, then a(t) = w(t) and we have z(t) ∈ ϕ( p, t). For otherwise, we may assume that an (t) ∈ D( pn , t) for all n ≥ 1. By (3.4), we have pn · an (t) = pn · w(t) for all n ≥ 1 which implies p · a(t) = p · w(t). Since each an (t) maximizes u t on the budget set B pn (t), it follows that a(t) maximizes u t on the budget set B pn (t) and so a(t) ∈ D( p, t). It follows that z(t) ∈ ϕ( p, t). Thus, ϕ( p, t) is upper semicontinuous.  

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