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

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).

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

$$\begin{aligned} \alpha v_{1} + (1 - \alpha )v_{2} \in \psi (p)\quad \hbox {for any } v_{1}, v_{2} \in \psi (p) \hbox { and any } 0 \le \alpha \le 1. \end{aligned}$$

(4.1)

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

$$\begin{aligned} g(S) = \int _{S}(f_{1}(p, t), f_{2}(p, t))\,dt \end{aligned}$$

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

$$\begin{aligned} \int _{S}(f_{1}(p, t), f_{2}(p, t))\,dt= & {} \alpha \left[ \int _{T}(f_{1}(p, t), f_{2}(p, t))\,dt\right] + (1 - \alpha )0 \\= & {} \alpha \left[ \int _{T}(f_{1}(p, t), f_{2}(p, t))\,dt\right] . \end{aligned}$$

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

$$\begin{aligned} f''(p, t) = \left\{ \begin{array}{ll} f_{1}(p, t) &{} \quad \hbox {if } t \in S\\ f_{2}(p, t) &{} \quad \hbox {if } t \not \in S. \end{array} \right. \end{aligned}$$

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,

$$\begin{aligned} \int _{T}f''(p, t)\,dt= & {} \int _{S}f_{1}(p, t)\,dt + \int _{T {\setminus } S}f_{2}(p, t)\,dt \\= & {} \alpha \int _{T}f_{1}(p, t)\,dt + (1 - \alpha ) \int _{T}f_{2}(p, t)\,dt. \end{aligned}$$

It follows that

$$\begin{aligned} \alpha v_{1} + (1 - \alpha )v_{2}= & {} \alpha \left[ \int _{T}f_{1}(p, t)\,dt - \int _{T}\mathbf{w}(t)\,dt\right] \\&+ (1 - \alpha )\left[ \int _{T}f_{2}(p, t)\,dt - \int _{T}\mathbf{w}(t)\,dt\right] \\= & {} \alpha \int _{T}f_{1}(p, t)\,dt + (1 - \alpha ) \int _{T}f_{2}(p, t)\,dt - \int _{T}\mathbf{w}(t)\,dt \\= & {} \int _{T}f''(p, t)\,dt - \int _{T}\mathbf{w}(t)\,dt \in \psi (p). \end{aligned}$$

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

$$\begin{aligned} \{(x, t)|x \in \varphi (p, t)\}= & {} [(F(p, t) \cap K)\cup (\hat{K} \cap \{\lambda d : d \in F(p, t), 0 \le \lambda < 1\})]\\&\times [\{t : u_{t}(D(p, t)) > u_{t}(\mathbf{x}(t))\} \cup \{t : u_{t}(D(p, t))= u_{t}(\mathbf{x}(t))\} \\&\cup \{t : u_{t}(D(p, t)) < u_{t}(\mathbf{x}(t))\}]\\= & {} [(F(p, t) \cap K)\cup (\hat{K} \cap \{\lambda d : d \in F(p, t), 0 \le \lambda < 1\})] \\&\times \, \{t : u_{t}(D(p, t)) > u_{t}(\mathbf{x}(t))\} \\&\cup \, [(F(p, t) \cap K)\cup (\hat{K} \cap \{\lambda d : d \in F(p, t), 0 \le \lambda < 1\})] \\&\times \, \{t : u_{t}(D(p, t)) = u_{t}(\mathbf{x}(t))\}\\&\cup \, [(F(p, t) \cap K)\cup (\hat{K} \cap \{\lambda d : d \in F(p, t), 0 \le \lambda < 1\})] \\&\times \, \{t : u_{t}(D(p, t)) < u_{t}(\mathbf{x}(t))\}\\= & {} [(F(p, t) \cap K)\cup (\hat{K} \cap \{\lambda d : d \in F(p, t), \lambda \ge 0\})] \\&\times \, \{t : u_{t}(D(p, t)) > u_{t}(\mathbf{x}(t))\}\\&\cup \, [(F(p, t) \cap K)\cup (\hat{K} \cap \{\lambda d : d \in F(p, t), \lambda \ge 0\})]\\&\times \,\{t : u_{t}(D(p, t)) = u_{t}(\mathbf{x}(t))\}\\&\cup \, [(F(p, t) \cap K)\cup (\hat{K} \cap \{\lambda d : d \in F(p, t), \lambda \ge 0\})]\\&\times \, \{t : u_{t}(D(p, t)) < u_{t}(\mathbf{x}(t))\}. \end{aligned}$$

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

$$\begin{aligned} z_{n}(t) = \lambda _{n}(t)a_{n}(t). \end{aligned}$$

(4.2)

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

$$\begin{aligned} z(t) = \lambda (t)a(t). \end{aligned}$$

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

$$\begin{aligned} p \cdot a(t) = p \cdot \mathbf{w}(t). \end{aligned}$$

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. $\square$