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Risk sharing and retrading in incomplete markets

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Abstract

At a competitive equilibrium of an incomplete-markets economy agents’ marginal valuations for the tradable assets are equalized ex-ante. We characterize the finest partition of the state space conditional on which this equality holds for any economy. This leads naturally to a necessary and sufficient condition on information that would induce agents to retrade, if such information was to become publicly available after the initial round of trade.

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Notes

  1. Here, and in what follows, we denote by \({u^h}^{\prime }\) and \({u^h}^{\prime \prime }\) the first and second derivative of the utility function \(u^h\).

  2. The same partition is employed by (Geanakoplos and Mas-Colell (1989), Section III) in order to characterize the degree of indeterminacy of equilibria with nominal assets.

  3. The partition \(\mathcal S (R)\) can be calculated by studying the equation system \(\mathcal L (a):=\sum _{s\in S}a_sr_s=0\), where \(a:=\{a_s\}\) is a vector in \(\mathbb R ^S\). A subset \(\hat{S}\) of \(S\) is a union of equal-valuation events if and only \(\sum _{s\in \hat{S}}a_sr_s=0\), for every zero of \(\mathcal L \). By checking this condition for every subset of \(S\), we can determine the partition \(\mathcal S (R)\).

  4. More precisely, given a subset \(E\) of Euclidean space, endowed with the relative Euclidean topology, we say that \(E^{\prime }\subset E\) is a generic subset of \(E\) if it is open and dense in \(E\).

  5. These definitions reduce to the usual notions of completeness and incompleteness if all the rows of \(R\) are nonzero.

  6. If \(S_k=J_k\), the fact that \(\mathrm{rank}(R_k)=J_k\) implies that \(R_k\) is column-equivalent to the identity matrix, so that \(S_k\) is not an equal-valuation event unless it is trivial.

  7. It is also worth noting that if \(R\) is in general position, so is any \(R^{\prime }\) that is column-equivalent to \(R\).

  8. The assumption that information arrives before date 0 consumption is essentially just an analytical convenience. In our setup, retrade occurs when the marginal rates of substitution between assets and date 0 consumption are not equal for a pair of agents. If information arrives after date 0 consumption, we can replace this by the equivalent condition that the marginal rate of substitution between a pair of assets is not equal for a pair of agents.

  9. We provide an example of such a signal in Example 3 at the end of this section.

  10. Notice that since \(\varPi \) is not an open subset of \(\mathbb R ^{S\varSigma }\), a generic subset of \(\varPi \) is open in \(\varPi \) but not necessarily open in \(\mathbb R ^{S\varSigma }\) (an open subset of \(\varPi \) is the intersection of \(\varPi \) with an open subset of \(\mathbb R ^{S\varSigma }\); see footnote 4). In particular, a generic subset of \(\varPi \) may include public signals that induce a partition of \(S\) and hence lie on the boundary of \(\varPi \).

  11. A special case of this result, when \(R\) is in general position (so that there is only one equal-valuation event), can be found in Gottardi and Rahi (2012).

  12. We choose to state Theorem 4.1 for a generic set of endowments that is a subset of \(\hat{\varOmega }\), even though this is not required by our argument, in order to facilitate comparison with our other results.

  13. The rank condition in the definition of \(\hat{\varPi }_2\) allows for the possibility that \(\{\pi _{s|\sigma }\}_{s\in S_k}\) is proportional to \(\{\overline{\pi }_s\}_{s\in S_k}\) for some values of \(\sigma \).

References

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Correspondence to Rohit Rahi.

Appendix

Appendix

Proof of Lemma 3.1

Suppose there are distinct partitions \(\{S^{\prime }_k\}_{k\in K^{\prime }}\) and \(\{S^{\prime \prime }_k\}_{k\in K^{\prime \prime }}\) such that the subspaces \(\{L^{\prime }_k\}\) corresponding to \(\{S^{\prime }_k\}\) are linearly independent, as are the subspaces \(\{L^{\prime \prime }_k\}\) corresponding to \(\{S^{\prime \prime }_k\}\). It suffices to show that the join of \(\{S^{\prime }_k\}\) and \(\{S^{\prime \prime }_k\}\), that is, the coarsest partition contained in both \(\{S^{\prime }_k\}\) and \(\{S^{\prime \prime }_k\}\), which we denote by \(\{\bar{S}_k\}_{k\in \bar{K}}\), also has the property that the subspaces \(\{\bar{L}_k\}\) corresponding to it are linearly independent.

Consider \(\bar{\ell }_k\in \bar{L}_k\) such that \(\sum _{k\in \bar{K}} \bar{\ell }_k=0\). Since \(\{\bar{S}_k\}\) is a refinement of \(\{S^{\prime }_k\}\), we can group the elements of \(\{\bar{S}_k\}_{k\in \bar{K}}\) so that the cells \(\{\bar{S}_k\}_{k\in {K_1}}\) are in \(S^{\prime }_1\), the cells \(\{\bar{S}_k\}_{k\in {K_2}}\) are in \(S^{\prime }_2\), and so on, with \(\cup _i K_i=\bar{K}\). Thus, we can write \(\sum _{k\in \bar{K}} \bar{\ell }_k=\sum _i\sum _{k\in K_i}\bar{\ell }_k=0\). For each \(i\), \(\sum _{k\in K_i}\bar{\ell }_k\) is in \(L^{\prime }_i\) and is equal to zero by the linear independence of \(\{L^{\prime }_i\}_{i\in K^{\prime }}\). Moreover, each element of the sum \(\sum _{k\in K_i}\bar{\ell }_k\) belongs to a distinct subspace \(L^{\prime \prime }_j\) and hence is equal to zero by the linear independence of \(\{L^{\prime \prime }_j\}_{j\in K^{\prime \prime }}\). Therefore, \(\bar{\ell }_k=0\) for all \(k\in \bar{K}\), that is, the subspaces \(\{\bar{L}_k\}_{k\in \bar{K}}\) are linearly independent. \(\square \)

Proof of Lemma 3.2

The matrices \(R\) and \(R^{\prime }\) are column-equivalent if and only if \(R^{\prime }=RX\), for some \(J\times J\) nonsingular matrix \(X\). Let \(\mathcal S (R)=\{S_1,\ldots , S_K\}\) be the equal-valuation partition for \(R\), and let \(\bar{R}_k\) be the \(S_k\times J\) submatrix of \(R\) consisting of the rows of \(R\) corresponding to the states in \(S_k\). Similarly, let \(\bar{R}^{\prime }_k\) be the \(S_k\times J\) submatrix of \(R^{\prime }\) corresponding to \(S_k\). Consider a vector \(a\in \mathbb R ^S\), and let \(a_k\in \mathbb R ^{S_k}\) be the elements of \(a\) corresponding to \(S_k\). We have \(a^{\top }R^{\prime }=a^{\top }RX\) and \(a_k^{\top }\bar{R}^{\prime }_k=a_k^{\top }\bar{R}_kX\).

Now, suppose \(a^{\top }R^{\prime }=0\). Then, \(a^{\top }R=\sum _{k\in K}a_k^{\top }\bar{R}_k=0\). Since the subspaces \(\{L_k\}\) are linearly independent, we must have \(a_k^{\top }\bar{R}_k=0\), for all \(k\). It follows that \(a_k^{\top }\bar{R}^{\prime }_k=0\), for all \(k\), and hence, the subspaces \(\{L^{\prime }_k\}\) are linearly independent. Moreover, since \(\{L_k\}\) is a maximal set of linearly independent subspaces, so is \(\{L^{\prime }_k\}\). This establishes that \(\mathcal S (R)=\mathcal S (R^{\prime })\).

Assuming for the moment that all the rows of \(R\) are nonzero, we now show that there exists a \(J\times J\) nonsingular matrix \(X\) such that \(RX\) can be written in the form \(\mathrm{diag}_{k\in K}[R_k]\) as asserted in the statement of the lemma. Let \(M_k\) be the \(J_k\)-dimensional subspace of \(\mathbb R ^J\) that is the orthogonal complement of the subspace generated by \(\{L_{\hat{k}}\}_{\hat{k}\ne k}\). We claim that the subspaces \(\{M_k\}\) are linearly independent. Indeed, consider \(m_k\in M_k\) such at \(\sum _k m_k = 0\). Then, \(\ell _k\cdot \sum _k m_k = 0\), for all \(\ell _k\in L_k\). But \(\ell _k\cdot m_{\hat{k}} = 0\), for all \(\hat{k}\ne k\). Therefore, \(\ell _k\cdot \sum _k m_k=\ell _k\cdot m_k = 0\), for all \(\ell _k\in L_k\), that is, \(m_k\) is orthogonal to \(L_k\). By the definition of \(M_k\), \(m_k\) is orthogonal to \(L_{\hat{k}}\), for all \(\hat{k}\ne k\). Consequently, \(m_k\) is orthogonal to \(\mathbb R ^J\), implying that it is zero. The same argument applies for all values of \(k\).

Let \(X_k\) be a \(J\times J_k\) matrix whose columns are a basis of \(M_k\). Thus, every column of \(X_k\) is orthogonal to every row of \(R\) that does not correspond to the states in \(S_k\). Therefore, \(\bar{R}_{\hat{k}}X_k=0\), for all \(\hat{k}\ne k\). Let \(X:=(X_1 \ldots X_K)\). Then, \(RX=\mathrm{diag}_{k}[R_k]\), where \(R_k:= \bar{R}_kX_k\), an \(S_k\times J_k\) matrix. Since the subspaces \(\{M_k\}\) are linearly independent, \(X\) is nonsingular. This proves that \(R\) is column-equivalent to \(\mathrm{diag}_{k\in K}[R_k]\). Moreover, \(\mathrm{rank}(R_k)=\mathrm{rank}(\bar{R}_k)=J_k\).

In the foregoing proof, the rows of \(R\) were assumed to be nonzero. If there are some zero rows, the same argument can be applied to the matrix \(R^{\star }\) and the set \(K^{\star }\) to show that \(R^{\star }\) is column-equivalent to \(\mathrm{diag}_{k\in K^{\star }}[R_k]\), and therefore, \(R\) is column-equivalent to (3). \(\square \)

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Gottardi, P., Rahi, R. Risk sharing and retrading in incomplete markets. Econ Theory 54, 287–304 (2013). https://doi.org/10.1007/s00199-012-0717-z

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