A limit result on bargaining sets

Abstract

We introduce a notion of bargaining set for finite economies and show its convergence to the set of Walrasian allocations.

Appendix

Proof of Proposition 3.1

Let (S, y) be a Walrasian objection to x. Assume that it is counter-objected in some replicated economy $r\mathcal{E}.$ That is, there exist $T \subseteq N$ and natural numbers $r_i \le r$ for each $i \in T,$ such that: $\sum _{i\in T} r_i z_i \le \sum _{i\in T} r_i \omega _i; \ z_i \succ _i y_i$ for every $i\in T \cap S$ and $z_i \succ _i x_i$ for every $i\in T {\setminus } S.$ Since (S, y) is a Walrasian objection at prices p we have that $p \cdot z_i>p \cdot \omega _i,$ for every $i\in T \cap S$ and $p \cdot z_i>p \cdot \omega _i,$ for every $i\in T {\setminus } S.$ This implies $p \cdot \sum _{i\in T} r_i z_i > p \cdot \sum _{i\in T} r_i \omega _i,$ which is a contradiction. Thus, we conclude that (S, y) is a justified$^{*}$ objection.

To show the converse, let (S, y) be a justified$^{*}$ objection to x and let $a=(a_1, \ldots , a_n)$ be an allocation such that $a_i=y_i$ if $i\in S$ and $a_i=x_i$ if $i\notin S.$ For every i define $\Gamma _{i}= \{ z \in {\mathbb {R}}^{\ell } | z+ \omega _{i} \succsim _{i} a_{i} \} \bigcup \{0\}$ and let $\Gamma$ be the convex hull of the union of the sets $\Gamma _{i}, i \in N.$ A similar proof to the limit theorem on the core by Debreu and Scarf (1963) shows that $\Gamma \bigcap (- {\mathbb {R}}^{\ell }_{++})$ is empty, which implies that 0 is a frontier point of $\Gamma .$ Then, there exists a price system p such that $p \cdot z \ge 0$ for every $z \in \Gamma .$ Therefore, we conclude that (S, y) is a Walrasian objection. $\square$

To prove Theorem 4.1 we show the following lemma.

Lemma

Let x be a non-Walrasian feasible allocation in the economy $\mathcal{E}.$ Then, the following statements hold:

• (i) For each i,  there exist a sequence of rational numbers $r^{k}_{i} \in (0,1]$ converging to 1 and a sequence of allocations $(x^{k}, k \in {\mathbb {N}})$ that converges to x such that: (a) $\sum _{i=1}^{n} r^{k}_{i} x^{k}_{i} \le \sum _{i=1}^{n} r^{k}_{i} \omega _{i},$ (b) $x^{k}_{i} \succ _{i} x_{i}$ for every i, sl and (c) $x^{k}_{i} \succ _{i} x^{k+1}_{i}$ for every k and every i.

Let $r^{k}= \sum _{i \in N} r^{k}_{i}$ and $\alpha ^{k}= (r^{k}_{i}/r^{k}, i \in N) \in (0,1]^{n}.$ Let $f^{k}$ be the step function given by $f^{k}(t) = x^{k}_{i}$ for every $t \in I_{i}(\alpha ^{k})$ in the continuum economy $\mathcal{E}_{c}(\alpha ^{k}).$

• (ii) If rx belongs to $\mathcal{B}^{*}(r\mathcal{E})$ for every replicated economy, then for every k,  there is a justified objection $(S^{k}, g^{k}),$ in the sense of Mas-Colell, to $f^{k}$ in the economy $\mathcal{E}_{c}(\alpha ^{k})$.

Proof of (i)

Observe that if $x^{k}$ converges to x and $x^{k}_{i} \succ _{i}x_{i}$, for every i and k,  then under continuity of preferences, condition (c) holds by taking a subsequence if necessary.

If x is a feasible allocation that is not efficient, then, for every i,  there exists $y_i$ such that $\sum _{i=1}^{n} y_i \le \sum _{i=1}^{n} \omega _i$ and $y_i \succ _{i} x_i.$ The sequence given by $x_i^k = \frac{1}{k}y_i + (1- \frac{1}{k})x_i$ fulfills the requirements in (a) with $r^{k}_{i}=1$ for all i and k.

Let x be a non-Walrasian feasible allocation which is efficient. Then, there exist rational numbers $a_{i} \in (0,1]$ (with $a_j < 1$ for some j) and bundles $y_{i}$ for all $i=1,\dots , n,$ such that $\sum _{i =1}^n a_{i} (y_{i} - \omega _{i}) = - \delta ,$ with $\delta \in {\mathbb {R}}^{\ell }_{++}$ and $y_{i} \succ _{i} x_{i},$ for every i (see Hervés-Beloso and Moreno-García 2001, for details). Let $a=\sum _{i=1}^n a_{i}.$ Given $\varepsilon \in (0,1],$ let $y_{i}^{\varepsilon } = \varepsilon y_{i} + (1- \varepsilon ) x_{i}.$ By convexity of preferences, $y_{i}^{\varepsilon } \succ _{i} x_{i}$ for every i. Consider $x_{i}^{\varepsilon }= x_{i} + \frac{\varepsilon \delta }{a^{\varepsilon }},$ where $a^{\varepsilon }= (1 - \varepsilon ) (n- a ).$ By monotonicity, $x^{\varepsilon }_{i} \succ _{i} x_{i}$ for every i. Take a sequence of rational numbers ${\varepsilon _{k}}$ converging to zero and, for each k and i,  let $a_{i}^{k} = (1- \varepsilon _{k})(1- a_{i})$, $r^{k}_{i}= a_{i} + a_{i}^{k} \in (0,1],$ and define $x^{k}_{i}= \frac{a_{i}}{r^{k}_{i}} y^{\varepsilon _{k}}_i + \frac{a^{k}_{i}}{r^{k}_{i}} x_{i}^{\varepsilon _{k}}.$ By construction, the sequences $r^{k}_{i}$ and $x^{k}_i$ ($i=1,\dots , n$ and $k \in {\mathbb {N}})$ verify the required properties.

Proof of (ii)

Let $q^{k}$ be a natural number such that $r^{k}_{i}= b^{k}_{i}/q^{k},$ with $b^{k}_{i} \in {\mathbb {N}}$ for each i. Since $x \in \bigcap _{r \in {\mathbb {N}}} \hat{\mathcal{B}}(r\mathcal{E}),$ $x^{k}$ cannot be a Walrasian allocation for the economy formed by $b^{k}_{i}$ agents of type i;  otherwise, the coalition formed by $b^{k}_{i}$ members of each type i joint with $x^{k}$ would define a justified$^{*}$ objection in the $q^{k}$-replicated economy.¹⁸ Then, $f^{k}$ cannot be a competitive allocation in $\mathcal{E}_{c}(\alpha ^{k}).$ By Mas-Colell’s (1989) equivalence result, $f^{k}$ is blocked by a justified objection $(S^{k}, g^{k})$ in $\mathcal{E}_{c}(\alpha ^{k}).$ By convexity of preferences, we can consider without loss of generality that $g^{k}$ is an equal treatment allocation. $\square$

Proof Theorem 4.1

Since $W(\mathcal{E}) \subseteq C(r\mathcal{E}),$ it is immediate that $W(\mathcal{E}) \subseteq \bigcap _{r \in {\mathbb {N}}} \hat{\mathcal{B}}(r\mathcal{E}).$

To show the converse, assume that x is a non-Walrasian allocation that belongs to $\bigcap _{r \in {\mathbb {N}}} \hat{\mathcal{B}}(r\mathcal{E}).$ Consider the sequence of justified objections $(S^{k}, g^{k})$ to $f^{k}$ in the economy $\mathcal{E}_{c}(\alpha ^{k})$ as constructed in the previous lemma. Let $\gamma ^{k}= \left( \gamma ^{k}_{i}= \mu (S^{k} \cap I_{i}(\alpha ^{k}))/\right. \left. \mu (S^{k}), i \in N \right) \in [0,1]^{n}.$ Since the number of types of consumers is finite, without loss of generality we can consider, taking a subsequence if necessary, that $N_{\gamma ^{k}}= \{ i \in N | \gamma ^{k}_{i} >0\}=T$ for every k. We use the same notation for such a subsequence and write $\gamma ^k_i$ converges to $\gamma _{i}$ for every $i \in T$ and $\sum _{i \in T} \gamma _{i}=1.$ Consider the sequence of economies $\mathcal{E}_{c}(\gamma ^{k})$ and the limit vector $\gamma .$

Then, by the previous lemma, for each natural number k,  there is a subset T of types and a competitive equilibrium $(p^{k}, g^{k})$ in $\mathcal{E}_{c}(\gamma ^{k})$ such that:

• (i) $g^{k}_{i} \succsim _{i} x^{k}_{i}$ for every $i \in T,$ with $g^{k}_{j} \succ _{j} x^{k}_{j}$ for some $j \in T,$ and

• (ii) $g^{k}_{i} \in d_{i}(p^{k})$ for every $i \in T,$ and $x_{i}^{k} \succsim _{i} d_{i}(p^{k})$ for every $i \in N {\setminus } T$.¹⁹

Let $A_{k}=\left\{ i \notin T | x_{i} \succsim _{i} d_{i}(p^{k}) \right\} ,$ $B_{k}= \left\{ i \notin T | x_{i} \prec _{i} d_{i}(p^{k}) \right\} .$ Since the number of types is finite, without loss of generality we can consider, taking a subsequence if it is necessary, that $A_{k}=A$ and $B_{k}=B$ for every k.

Let us choose a sequence of numbers $\delta _{k} \in (0,1)$ converging to 1, and let $\varepsilon ^{k}= 1 - \delta _{k}$, which converges to zero. For each $i \in B$ take $\varepsilon ^{k}_{i}>0$ such that $\varepsilon ^{k}= \sum _{i \in B} \varepsilon ^{k}_{i}.$ Let $T_{1}=T\cup B$, and for each $i \in T_{1}$ define $\widetilde{\gamma }^{k}_{i} \in (0,1)$ as follows:

$$\begin{aligned} \widetilde{\gamma }^{k}_{i}= \left\{ \begin{array}{cl} \delta _{k} \gamma ^{k}_{i} &{} \hbox { if } i \in T \\ \varepsilon ^{k}_{i} &{} \hbox { if } i \in B \end{array} \right. \end{aligned}$$

Note that $\sum _{i \in T_1} \widetilde{\gamma }^{k}_{i}=1.$ Moreover, $\lim _{k \rightarrow \infty } \widetilde{\gamma }^{k}_{i}= \lim _{k \rightarrow \infty } \gamma ^{k}_{i} = \gamma _{i}$ for every $i \in T$ and $\widetilde{\gamma }^{k}_{i}$ goes to zero as k increases for every $i \in B.$ Then, the economy $\mathcal{E}_{c}(\widetilde{\gamma }^{k})$ differs from $\mathcal{E}_{c}(\gamma ^{k})$ only in at most a finite set of types of agents whose measure goes to zero when k increases. Now, for each k and for each $i \in T_{1}=T\cup B,$ take a sequence of positive rational numbers $\gamma ^{km}_{i}$ converging to $\widetilde{\gamma }^{k}_{i}$ when m increases and such that $\sum _{i \in T_1} \gamma ^{km}_{i} =1$ for every m. In this way, for each k,  let us consider the sequence of continuum economies $\mathcal{E}_{c}(\gamma ^{km}).$ To simplify notation, let $\mathcal{E}^{kk}_{c}= \mathcal{E}_{c}(\gamma ^{kk}).$ Note that $\lim _{k \rightarrow \infty } \gamma ^{kk}_{i} = \lim _{k \rightarrow \infty } \gamma ^{k}_{i}$ for every $i \in T$ and $\lim _{k \rightarrow \infty } \gamma ^{kk}_{i}=0$ for every $i \in B.$ Then, the sequence $\gamma ^{kk}$ that describes the diagonal sequence of economies $\mathcal{E}^{kk}_{c}$ converges to $\gamma$ as well.

Then, by the continuity of the equilibrium correspondence at $\gamma$ and the continuity of preferences, we deduce that for every k large enough there is an equilibrium price $\widetilde{p}^{k}_{1}$ for the economy $\mathcal{E}^{kk}_{c}$ such that $d_{i}(\widetilde{p}^{k}_{1}) \succ _{i} x _{i}$ for every $i \in T_{1}.$ If $x_{i} \succsim _{i} d_{i}(\widetilde{p}^{k}_{1})$ for every $i \in A,$ we have found a Walrasian objection to x in a replicated economy, which is in contradiction to the fact that x belongs to $\bigcap _{r \in {\mathbb {N}}} \hat{\mathcal{B}}(r\mathcal{E}).$ Otherwise, let $\widetilde{A}_{k}= \left\{ i \notin T_{1} | x_{i} \succsim _{i} d_{i}(\widetilde{p}^{k}_{1}) \right\} ,$ $\widetilde{B}_{k}= \left\{ i \notin T_{1} | x_{i} \prec _{i} d_{i}(\widetilde{p}^{k}_{1}) \right\} .$ As before, without loss of generality, taking a subsequence if it is necessary, we can consider $\widetilde{A}_{k} = \widetilde{A}$ and $\widetilde{B}_{k}= \widetilde{B}$ for every k. Let $T_{2}= T_{1} \cup \widetilde{B}$ and repeat the analogous argument. In this way, after a finite number h of iterations, we have either (i) $T_{h}= N =\{1,\ldots , n\}$ or (ii) $N {\setminus } T_{h} \ne \emptyset$ but $\left\{ i \notin T_{h} | x_{i} \prec _{i} d_{i}(\widetilde{p}^{k}_{h}) \right\} = \emptyset .$ If (i) occurs we find a justified$^{*}$ objection to x in a replicated economy which involves all the types of agents. If (ii) is the case, there is also a justified$^{*}$ objection to x in a replicated economy but involving only a strict subset of types. In both cases we obtain a contradiction. $\square$