Coincidence of the Mas-Colell bargaining set and the set of competitive equilibria in a continuum coalition production economy

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).

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.

Authors and Affiliations

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

   Jiuqiang Liu

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

   Jiuqiang Liu

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

   Huihui Zhang

Corresponding author

Correspondence to Jiuqiang Liu.

Appendix

We first state the well-known Lyapunov Theorem which is needed for the next lemma.

Lyapunov Theorem

Let \((T, {\mathcal {F}}, \mu )\) be an atomless finite measure space and \(\mathbf{f}\) is an integrable function from T into \({\mathbb {R}}^{l}\). Then the set \(\{\int _{S}\mathbf{f}\,d\mu : S \in {\mathcal {F}}\}\) is a convex set in \({\mathbb {R}}^{l}\).

Lemma 4.1

The correspondence \(\psi (p)\) is convex for each \(p \in P\).

Proof

Fix \(p \in P\). To show \(\psi (p)\) is convex, we need to show that

Let \(v_{1}\), \(v_{2}\) be any two members in \(\psi (p)\) and \(\alpha \) be any constant satisfying \(0 \le \alpha \le 1\). By the definition of \(\psi (p)\), there exist \(f_{1}(p, t), f_{2}(p, t) \in \varphi (p, t)\) such that \(v_{1} = \int _{T}f_{1}(p, t)\,dt - \int _{T}\mathbf{w}(t)\,dt\) and \(v_{2} = \int _{T}f_{2}(p, t)\,dt - \int _{T}\mathbf{w}(t)\,dt\). Let \(g : {\mathcal {F}} \mapsto {\mathbb {R}}^{2l}\) be the atomless measure given by

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

which implies \(\int _{S}f_{1}(p, t)\,dt = \alpha \int _{T}f_{1}(p, t)\,dt\) and \(\int _{T {\setminus } S}f_{2}(p, t)\,dt = (1 - \alpha ) \int _{T}f_{2}(p, t)\,dt\). Define

Then \(f''(p, t) \in \varphi (p, t)\) as \(f_{1}(p, t), f_{2}(p, t) \in \varphi (p, t)\) and so \(\int _{T}f''(p, t)\,dt - \int _{T}\mathbf{w}(t)\,dt \in \psi (p)\). Moreover,

It follows that

Thus, (4.1) holds and \(\psi (p)\) is convex for each \(p \in P\). \(\square \)

Let F be a set-valued function from T to \({\mathbb {R}}_{+}^{l}\). F is called Borel measurable if its graph, \(\{(x, t) | x \in F(t)\}\), is a Borel subset of \({\mathbb {R}}_{+}^{l} \times T\).

Lemma 4.2

For fixed \(p \in P,\) the correspondence \(\varphi (p, t)\) is Borel measurable.

Proof

Fix \(p \in P\). We need to prove that \(\{(x, t)|x \in \varphi (p, t)\}\) is a Borel subset of \({\mathbb {R}}_{+}^{l} \times T\). Note from the definitions that if \(F(p, t)) \cap K = \emptyset \), then \(\sum _{i = 1}^{l}d^{j} > \alpha \) for any \(d \in F(p, t)\) which implies that \(\hat{K} \cap \{\lambda d : d \in F(p, t), \lambda \ge 1\} = \emptyset \). It follows from the definition of \(\varphi (p, t)\) that

Recall that K and \(\hat{K}\) are closed and F(p, t) is closed for each \(p \in P\) and each \(t \in T\), the set \((F(p, t) \cap K)\cup (\hat{K} \cap \{\lambda d : d \in F(p, t), \lambda \ge 0\})\) is closed and thus Borel. By the measurability assumption (A.4), the sets \(\{t : u_{t}(D(p, t)) > u_{t}(\mathbf{x}(t))\}\), \(\{t : u_{t}(D(p, t)) = u_{t}(\mathbf{x}(t))\}\), and \(\{t : u_{t}(D(p, t)) < u_{t}(\mathbf{x}(t))\}\) are Borel subsets. Thus, \(\{(x, t)|x \in \varphi (p, t)\}\) is a Borel subset of \({\mathbb {R}}_{+}^{l} \times T\). \(\square \)

We need the following result from Aumann (1965) for the proof of the next lemma.

Proposition 4.3

(Aumann 1965) Let \(F_{x}(t)\) be a set-valued function defined for \(t \in T\) and \(x \in A,\) all of whose values are bounded by the same integrable point-valued function,  and such that \(F_{x}\) is Borel measurable for each fixed \(x \in A\). If \(F_{x}(t)\) is upper semicontinuous in x for each fixed t,  then \(\int _{T}F_{x}(t)\,dt\) is upper semicontinuous.

Lemma 4.4

The correspondence \(\psi (p)\) is upper semicontinuous for each \(p \in P\).

Proof

By Lemma 4.2 and Proposition 4.3, we only need to prove that the correspondence \(\varphi (p, t)\) is upper semicontinuous for each fixed \(t \in T\). Fix \(t \in T\). Let \(p_{n} \rightarrow p\) with all \(p_{n}, p \in P\), \(z_{n}(t) \rightarrow z(t)\) with \(z_{n}(t) \in \varphi (p_{n}, t)\) for all \(n \ge 1\). We need to prove that \(z(t) \in \varphi (p, t)\). For each \(n \ge 1\), since \(z_{n}(t) \in \varphi (p_{n}, t)\), there exists \(a_{n}(t) \in F(p_{n}, t)\) and \(0 \le \lambda _{n}(t) \le 1\) such that

By the definition of \(F(p_{n}, t)\), we have either \(a_{n}(t) = \mathbf{w}(t)\) or \(a_{n}(t) \in D(p_{n}, t)\) for each \(n \ge 1\). Since \(0 \le \lambda _{n}(t) \le 1\) for all \(n \ge 1\), \(\{\lambda _{n}(t)\}_{n \ge 1}\) has a convergent subsequence, say, \(\lambda _{n}(t) \rightarrow \lambda (t)\). Clearly, \(0 \le \lambda (t) \le 1\). It follows from (4.2) that \(a_{n}(t) \rightarrow a(t) \in {\mathbb {R}}_{+}^{l}\) and

If \(a_{n}(t) = \mathbf{w}(t)\) for infinitely many n, then \(a(t) = \mathbf{w}(t)\) and we have \(z(t) \in \varphi (p, t)\). For otherwise, we may assume that \(a_{n}(t) \in D(p_{n}, t)\) for all \(n \ge 1\). By (3.4), we have \( p_{n} \cdot a_{n}(t) = p_{n} \cdot \mathbf{w}(t)\) for all \(n \ge 1\) which implies

Since each \(a_{n}(t)\) maximizes \(u_t\) on the budget set \(B_{p_{n}}(t)\), it follows that a(t) maximizes \(u_t\) on the budget set \(B_{p_{n}}(t)\) and so \(a(t) \in D(p, t)\). It follows that \(z(t) \in \varphi (p, t)\). Thus, \(\varphi (p, t)\) is upper semicontinuous. \(\square \)

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