A limit result on bargaining sets


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

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  1. 1.

    Maschler (1976) discussed the advantages that the bargaining set has over the core.

  2. 2.

    The convexity of preferences we require is the following: If a consumption bundle z is strictly preferred to \(\hat{z}\) so is the convex combination \(\lambda z + (1-\lambda ) \hat{z}\) for any \(\lambda \in (0,1).\) This convexity property is weaker than strict convexity and it holds, for instance, when the utility functions are concave.

  3. 3.

    The preference relation \(\succsim _{i}\) is strictly monotone if \(x \ge y \), \(x\ne y\) implies that \(x \succ _i y\).

  4. 4.

    We wrote \({\mathbb {R}}^{\ell }_{++}\) for the sake of simplicity; it is actually enough that \(\omega _i \in {\mathbb {R}}^{\ell }_{+}\) as long as \(\omega = \sum _{i=1}^n \omega _i>>0\).

  5. 5.

    See García-Cutrín, and Hervés-Beloso (1993) for further details.

  6. 6.

    See Geanakoplos (1978), Mas-Colell (1989), Dutta et al. (1989), Zhou (1994) and Anderson (1998).

  7. 7.

    Note that this notion can be applied to any cooperative game. Note also that the bargaining set is non-empty since it contains the core.

  8. 8.

    Actually, this asymmetry is responsible for Lemma 3.2 in ATZ (1997).

  9. 9.

    Note that a justified objection as in ATZ in the economy \(r\mathcal{E}\) is not necessarily justified in the economy \((r+1) \mathcal{E}.\)

  10. 10.

    As it is stressed by Mas-Colell, a Walrasian objection requires a price system p and is based on a self-selection property. Within our approach, types that participate in a coalition in a Walrasian objection against an allocation are those who would trade at the price vector p rather than get the consumption bundle they receive by such an allocation.

  11. 11.

    This is so because if a coalition with a Mas-Colell justified objection includes only part of some types of agents, then it is not possible for these agents to strictly improve with the objection.

  12. 12.

    Note that \(\alpha =2\) defines the Walrasian allocation and \(V_{1}(1)= V_{2}(1)=4.\)

  13. 13.

    Recall that the set of economies on which the equilibrium correspondence is continuous is open and dense (see Hildenbrand 1972 or Dierker 1973).

  14. 14.

    These conditions refer basically to differentiability of the utility functions, degrees of risk aversion, elasticity of substitution for commodities and collinearity of endowments. See, for instance, Varian (1985), Mityushin and Polterovich (1978), Fisher (1972) and Mas-Colell (1991). See also Arrow and Hahn (1971), for additional details on uniqueness of equilibrium.

  15. 15.

    The standard example is a demand that comes from the maximization of a Cobb–Douglas utility function subject to a budget constraint with strictly positive endowments. A generalization is the utility function \(U(x)= \prod _{h=1}^{\ell } \left( x_{h}- \beta _{h}\right) ^{\gamma _{h}},\) with \(\beta _{h} \le 0, \ \gamma _{h}>0\) and \(\sum _{h=1}^{\ell } \gamma _{h}=1.\)

  16. 16.

    Note that this utility function is not differentiable.

  17. 17.

    For instance, we can take \(h_{1}=\left( \frac{11^2}{5^2 \left( 3-2^{1/4} \right) ^2}, \frac{11^2}{5^2 \left( 3-2^{1/4} \right) ^2} \right) \) and \(h_{2}=(3,3)-h_{1}.\)

  18. 18.

    We remark that any objecting coalition involving all types along with a Walrasian allocation for such a coalition defines a justified\(^{*}\) objection. This is not the case for the corresponding Mas-Colell’s notion.

  19. 19.

    Note that, given a price vector p,  all the bundles in \(d_{i}(p)\) are indifferent; thus, when we write \(z \succsim _{i} d_{i}(p)\) it means \(z \succsim _{i} d\) for every \(d \in d_{i}(p).\)


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Corresponding author

Correspondence to Emma Moreno-García.

Additional information

\(^{*}\) We are grateful to Carlos Hervés-Beloso for his helpful comments and suggestions. We are also indebted to Joao Correia-da-Silva and Marta Faias, the participants in the XXIII and XXIV European Workshop on General Equilibrium Theory and in the SAET 2015 Conference. We specially thank the comments and suggestions of an anonymous referee that helped to improve the final version of this work.

\(^{**}\) This work is partially supported by Research Grants ECO2016-75712-P (AEI/FEDER, UE), RGEA-ECOBAS (AGRUP2015/08 Xunta de Galicia) and SA072U16 (Junta de Castilla y León).



Proof of Proposition 3.1

Let (Sy) 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 (Sy) 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 (Sy) is a justified\(^{*}\) objection.

To show the converse, let (Sy) 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 (Sy) is a Walrasian objection. \(\square \)

To prove Theorem 4.1 we show the following lemma.


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

  1. (i)

    For each ithere 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 \(\mathcal{E}_{c}(\alpha ^{k}).\)

  1. (ii)

    If rx belongs to \(\mathcal{B}^{*}(r\mathcal{E})\) for every replicated economy, then for every kthere 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.Footnote 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:

  1. (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

  2. (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 \(i \in N {\setminus } T\).Footnote 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. \(\square \)

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Hervés-Estévez, J., Moreno-García, E. A limit result on bargaining sets. Econ Theory 66, 327–341 (2018). https://doi.org/10.1007/s00199-017-1063-y

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  • Bargaining set
  • Coalitions
  • Core
  • Veto mechanism
  • Justified objections

JEL Classification

  • D51
  • D11
  • D00