Economic Theory

, Volume 54, Issue 2, pp 287–304

Risk sharing and retrading in incomplete markets

Authors

  • Piero Gottardi
    • Department of EconomicsEuropean University Institute
    • Dipartimento di Scienze EconomicheUniversità di Venezia
    • Department of Finance and Financial Markets GroupLondon School of Economics
Research Article

DOI: 10.1007/s00199-012-0717-z

Cite this article as:
Gottardi, P. & Rahi, R. Econ Theory (2013) 54: 287. doi:10.1007/s00199-012-0717-z

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.

Keywords

Competitive equilibriumIncomplete marketsInformationRetrading

JEL Classification

D52D80

1 Introduction

Consider a two-period single-good economy with incomplete asset markets. It is well understood that competitive equilibria in this setting are constrained efficient in the sense that a Pareto improvement cannot be achieved by reallocating the existing assets (Diamond 1967), while being generically Pareto inefficient (see, for example, Magill and Quinzii 1996). In other words, at a competitive equilibrium, agents’ marginal valuations for the tradable assets are equal when evaluated ex-ante, but are typically not equal conditional on the true state of the world, for every realization of the uncertainty.

This suggests that information that partially resolves the uncertainty will typically induce agents to rebalance their portfolios, if such information was to become publicly available after the initial round of trade. We show that this is indeed the case and explicitly characterize the set of public signals that lead to retrade. Since retrading occurs after the arrival of information if and only if it generates disagreement among agents regarding the marginal value of assets, this characterization is closely tied to the events conditional on which asset valuations are equalized in equilibrium.

We define an equal-valuation event to be a subset of states \(\hat{S}\) with the property that, at a competitive equilibrium, agents’ marginal valuations for assets are equal conditional on \(\hat{S}\) for any economy, but generically not equal conditional on a strict subset of \(\hat{S}\). Thus, the set of equal-valuation events is the finest partition of the state space conditional on which agents’ asset valuations are equal in equilibrium for a generic economy. We show that information that affects only the relative probabilities of equal-valuation events does not lead to retrade while, for a generic subset of endowments, retrade does occur if the information alters the relative probabilities of states within an equal-valuation event. For a generic subset of endowments, therefore, the latter condition is both necessary and sufficient for the information to lead to retrade. If markets are incomplete, the subset of public signals that satisfy this condition is itself generic in the set of all public signals.

While there is a substantial literature on trading in financial markets in response to news, little has been said on the characteristics of news that induces agents to retrade. A class of no-trade results can be traced back to Milgrom and Stokey (1982) who show that the arrival of information does not lead to retrade if the initial allocation is Pareto efficient. This leaves open the question of retrading in a competitive economy with incomplete markets. In this setting, Blume et al. (2006) provide sufficient conditions on a public signal such that retrade occurs for a generic economy. We generalize their result (in Theorem 4.3) by providing a weaker sufficient condition, for a broader class of public signals (including signals that induce a partition of the state space) and for an arbitrary asset structure.

The paper is organized as follows. We describe the economy in the next section. In Sect. 3, we introduce the notion of an equal-valuation event and analyze its properties. Then, in Sect. 4, we consider a public signal observed by agents after the initial round of trade and characterize the set of signals that lead to retrade.

2 The economy

We consider an exchange economy under uncertainty with two periods, 0 and 1, and a single physical consumption good. Asset markets open at date 0. At date \(1\), assets pay off. Our aim is to identify the types of unanticipated public information that would lead to retrade, if such information was to arrive after agents have traded at date 0 but before the realization of the uncertainty. In this section, we describe the basic environment, with no arrival of information.

The economy is populated by \(H\ge 2\) agents, with typical agent \(h\in H\) (here, and elsewhere, we use the same symbol for a set and its cardinality). Uncertainty is parametrized by \(S\) states of the world. The probability of state \(s\) is \(\overline{\pi }_s\) (\(\overline{\pi }_s>0\) for all \(s\), and \(\sum _s\overline{\pi }_s=1\)). Agent \(h\in H\) has endowments \(\omega ^h_0 >0\) in period 0 and \(\omega ^h\in \mathbb R _{++}^{S}\) in period 1, and time-separable expected utility preferences with von Neumann-Morgenstern utility functions \(u_0^h:\mathbb R _{++}\rightarrow \mathbb R \) for period 0 consumption and \(u^h:\mathbb R _{++}\rightarrow \mathbb R \) for period 1 consumption. We assume that \(u^h\) is twice continuously differentiable, \({u^h}^{\prime }>0, {u^h}^{\prime \prime }<0\), and \(\lim _{c\rightarrow 0}{u^h}^{\prime }(c)=\infty \);1 the same assumptions apply to \(u^h_0\).

There are \(J\ge 2\) assets. Asset payoffs are given by the \(S\times J\) matrix \(R\), whose \((s,j)\)’th element is \(r^j_s\), the payoff of asset \(j\) in state \(s\). We denote the \(j\)’th column of \(R\) by \(r^j\) and the \(s\)’th row of \(R\) by \(r_s^{\top }\) (by default all vectors are column vectors, unless transposed). Thus, \(r^j\) is the vector of payoffs of asset \(j\), and \(r_s\) is the vector of asset payoffs in state \(s\). We assume, without loss of generality, that \(R\) has full column rank \(J\). Markets are complete if \(J=S\), and incomplete if \(J<S\).

We parametrize economies by agents’ date 1 endowments \(\omega :=\{\omega ^{h}\}_{h\in H} \in \varOmega :=\mathbb R _{++}^{SH}\). Let \(p\in \mathbb R ^{J}\) be the vector of asset prices (date 0 consumption serves as the numeraire), and \(y^{h}\in \mathbb R ^{J}\) the portfolio of agent \(h\). The consumption of agent \(h\) is then given by \(\omega _0^h-p\cdot y^h\) at date 0 and \(\omega _s^h+r_s\cdot y^h\) in state \(s\) at date 1. Let \(y:=\{y^{h}\}_{h\in H}\). A competitive equilibrium is defined as follows:

Definition 2.1

Given an economy \(\omega \in \varOmega \), a competitive equilibrium consists of a portfolio allocation \(y\) and prices \(p\), satisfying the following two conditions:
  1. (a)
    Agent optimization: \(\forall h\in H\), \(y^{h}\) solves
    $$\begin{aligned} \max _{y^h\in \mathbb R ^J}\, \left( u^h_0\!\left[\omega ^h_0 - p \cdot y^h \right] + \sum _{s\in S}\overline{\pi }_s \, u^{h}\!\left[\omega ^h_s + r_s \cdot y^h\right] \right). \end{aligned}$$
    (1)
     
  2. (b)
    Market clearing:
    $$\begin{aligned} \sum _{h\in H}y^h=0. \end{aligned}$$
    (2)
     

Notation

In our analysis, we use the following shorthand notation for matrices. Given an index set \(\mathcal{N}\) with typical element \(n\), and a collection \(\{z_n\}_{n\in \mathcal N }\) of vectors or matrices, we denote by \(\mathrm{diag}_{n\in \mathcal N }[z_n]\) the (block) diagonal matrix with typical entry \(z_n\), where \(n\) varies across all elements of \(\mathcal N \). In similar fashion, we write \([\ldots z_n \ldots _{n\in \mathcal N }]\) to denote the row block with typical element \(z_n\), and analogously for column blocks. We drop reference to the index set if it is obvious from the context: For example, \(\mathrm{diag}_{h\in H}\) is shortened to \(\mathrm{diag}_h\), and \([\ldots z_s \ldots _{s\in S}]\) to \([\ldots z_s \ldots _s]\). We use the same symbol \(0\) for the zero scalar and the zero matrix; in the latter case, we occasionally indicate the dimension in order to clarify the argument. A “\(*\)” stands for any term whose value is immaterial to the analysis.

3 Equal-valuation events and risk sharing

In our characterization of the set of public signals that lead to retrade, a key role is played by the notion of an equal-valuation event, a minimal event conditional on which agents’ asset valuations are equal in equilibrium. We formalize this notion as follows. Consider a partition of \(S\) given by \(\{S_1,\ldots ,S_K\}\). For each \(k\in K:=\{1,\ldots , K\}\), let \(L_k\) be the subspace of \(\mathbb R ^J\) spanned by the vectors \(\{r_s\}_{s\in S_k}\). We say that the subspaces \(L_1,\ldots ,L_K\) are linearly independent if \(\sum _{k\in K}\ell _k=0\), \(\ell _k\in L_k\), implies \(\ell _k=0\) for all \(k\). Henceforth, we choose a partition for which \(L_1,\ldots ,L_K\) are linearly independent, and \(K\) is maximal.2 We denote this partition by \(\mathcal S (R)\).3

Lemma 3.1

The partition \(\mathcal S (R)\) is unique.

The proof is in the Appendix. We will show below (Theorems 3.1 and 3.2) that an event \(S_k\) in \(\mathcal S (R)\) is a subset of \(S\) satisfying the two properties stated in the Introduction, namely that (a) conditional on this event, agents’ asset valuations are equal in equilibrium, and (b) conditional on a strict subset of this event, agents’ asset valuations are not equal at any equilibrium, for a generic (i.e., open and dense)4 subset of endowments. Thus, for a generic subset of endowments, \(\mathcal S (R)\) is the finest partition of \(S\) conditional on which asset valuations are equalized across agents in equilibrium. Accordingly, for a given asset payoff matrix \(R\), we refer to \(\mathcal S (R)\) as the equal-valuation partition and \(S_k\in \mathcal S (R)\) as an equal-valuation event.

If there is a state \(s\in S\) in which the payoff of every asset is zero, that is, \(r_s=0\), the singleton event \(\{s\}\) is clearly an equal-valuation event. It is useful to separate such zero-payoff states and denote by \(S^{\star }\) the set of states \(s\in S\) for which \(r_s\ne 0\). Without loss of generality, we can order the partition \(\mathcal S (R)\) so that the first \(K^{\star }\) equal-valuation events \(\{S_1,\ldots , S_{K^{\star }}\}\) are subsets of \(S^{\star }\), while the remaining equal-valuation events each consist of single zero-payoff state. Moreover, we can order the states in \(S\) so that the first \(S_1\) states correspond to the event \(S_1\), the following \(S_2\) states correspond to the event \(S_2\), and so on. Let \(R^{\star }\) be the submatrix of \(R\) consisting of the first \(S^{\star }\) rows (these are the nonzero rows of \(R\)), and let the dimension of \(L_k\) be denoted by \(J_k\). Then, we have \(\sum _{k\in K^{\star }}J_k=J\) (note that \(J_k=0\) for \(k\notin K^{\star }\)). The following lemma shows that the partition \(\mathcal S (R)\) is invariant to changes in asset payoffs that do not affect the column span of \(R\). Moreover, \(R^{\star }\) is column-equivalent to a block-diagonal matrix, with each block corresponding to an equal-valuation event \(S_k,\,k\in K^{\star }\):

Lemma 3.2

Suppose the asset payoff matrices \(R\) and \(R^{\prime }\) are column-equivalent. Then, \(\mathcal S (R)=\mathcal S (R^{\prime })\). Furthermore, \(R\) is column-equivalent to
$$\begin{aligned} \left(\frac{\mathrm{diag}_{k\in K^{\star }}[R_k]}{0_{(K-K^{\star })\times J}} \right), \end{aligned}$$
(3)
where \(R_k\) is an \(S_k \times J_k\) matrix with \(\mathrm{rank}(R_k)=J_k>0\).

The proof is in the Appendix. Lemma 3.2 implies that for each equal-valuation event that is not a zero-payoff state, a portfolio can be found that pays off only in that event. We say that an equal-valuation event \(S_k\) is trivial if it is a singleton and nontrivial otherwise. A trivial equal-valuation event consists of a single state that is either a zero-payoff state or an insurable state (i.e., for which the corresponding Arrow security can be replicated with the available assets), while a nontrivial equal-valuation event consists of two or more states, none of which is a zero-payoff state or an insurable state.

In our retrading results, an important role is played by the completeness or incompleteness of the asset structure, in particular relative to the nonzero-payoff states \(S^{\star }\). We say that asset markets are \(S^{\star }\)-complete if they are complete relative to \(S^{\star }\) (i.e., if \(J=S^{\star }\)), and \(S^{\star }\)-incomplete otherwise (if \(J<S^{\star })\).5 A nontrivial equal-valuation event exists if and only if markets are \(S^{\star }\)-incomplete. Moreover, an equal-valuation event \(S_k\) is nontrivial if and only if \(S_k>J_k>0\).6

Example 1

Suppose there are four states of the world: \(S=\{s_1,s_2,s_3,s_4\}\). Consider a nontraded cash flow that pays
$$\begin{aligned} d= \left(\begin{array}{c} 1 \\ 2 \\ 3 \\ 4 \end{array} \right). \end{aligned}$$
There are two traded assets, a debt claim on \(d\) with face value 2 and a residual equity claim. Thus, the asset payoff matrix is
$$\begin{aligned} R= \left(\begin{array}{cc} 1&\quad 0 \\ 2&\quad 0 \\ 2&\quad 1 \\ 2&\quad 2 \end{array} \right). \end{aligned}$$
It is easy to verify there is only one equal-valuation event, that is, \(\mathcal S (R)=\{S\}\). \(\quad \Vert \)

Example 2

Consider the asset structure in Example 1. Suppose that, in addition to risky debt and levered equity, a risk-free asset is also available. Thus, the asset payoff matrix is
$$\begin{aligned} R= \left(\begin{array}{ccc} 1&\quad 0&\quad 1\\ 2&\quad 0&\quad 1\\ 2&\quad 1&\quad 1\\ 2&\quad 2&\quad 1 \end{array} \right), \end{aligned}$$
which is column-equivalent to
$$\begin{aligned} \left(\begin{array}{ccc} 1&\quad 0&\quad 0\\ 0&\quad 1&\quad 0\\ 0&\quad 1&\quad 1\\ 0&\quad 1&\quad 2 \end{array} \right). \end{aligned}$$
Therefore, the equal-valuation partition \(\mathcal S (R)\) is given by \(\{S_1,S_2\}\), where \(S_1\) is a trivial equal-valuation event consisting of the single insurable state \(s_1\), and \(S_2=\{s_2,s_3,s_4\}\) is a nontrivial equal-valuation event. \(\quad \Vert \)

An assumption commonly employed in the incomplete-markets literature is that the asset payoff matrix \(R\) is in general position, meaning that every \(J \times J\) submatrix of \(R\) is nonsingular. If \(R\) is in general position, and markets are incomplete, there is only one equal-valuation event.7 The argument is as follows. Since all the rows of \(R\) are nonzero (due to the general position of \(R\)), and markets are incomplete, there exists a nontrivial equal-valuation event, which we can take to be \(S_1\) without loss of generality. By the general position property, any collection of \(J^{\prime }\) rows of \(R\), with \(J^{\prime }\le J\), is linearly independent, so that we must have \(S_1>J\). But this implies that \(\mathrm{dim}(L_1)=J\). Hence, there is no equal-valuation event other than \(S_1\). The converse is not true, however: The asset payoff matrix in Example 1 is not in general position; yet, there is only one equal-valuation event.

Henceforth, we will take \(R\) to be of the block-diagonal form (3). Due to Lemma 3.2, this entails no loss of generality.

We now characterize risk sharing at a competitive equilibrium in terms of equal-valuation events. The first-order conditions for the utility-maximization program (1),
$$\begin{aligned} \sum _{s\in S}\overline{\pi }_s\,{u^h}^{\prime }\!\left[\omega _s^h+r_{s}\cdot y^{h}\right]r_s-{u^h_0}^{\prime }\!\left[\omega _0^h-p\cdot y^h\right]p =0,\quad \forall h\in H, \end{aligned}$$
(4)
imply that
$$\begin{aligned} \frac{ \sum _{s\in S} \overline{\pi }_s\, {u^h}^{\prime }\left[\omega ^h_s+r_s\cdot y^h\right]\,r_s }{ {u_0^h}^{\prime }\left[\omega ^h_0-p\cdot y^h\right] } = \frac{ \sum _{s\in S} \overline{\pi }_s\, {u^{\hat{h}}}^{\prime }\left[\omega ^{\hat{h}}_s+r_s\cdot y^{\hat{h}}\right]\,r_s }{ {u_0^{\hat{h}}}^{\prime }\left[\omega ^{\hat{h}}_0-p\cdot y^{\hat{h}}\right] }, \quad \forall h,\hat{h}\in H.\quad \end{aligned}$$
(5)
Thus, asset valuations (by which we mean the marginal rates of substitution between assets and period 0 consumption) are equalized across agents when evaluated ex-ante, that is, at the time of trading. This is just the standard result that competitive equilibria are constrained Pareto efficient. In order to economize on notation, we let
$$\begin{aligned} \mu ^{h\hat{h}}(y,p):= \frac{{u_0^h}^{\prime }\left[\omega ^h_0-p\cdot y^h\right]}{{u_0^{\hat{h}}}^{\prime }\left[\omega ^{\hat{h}}_0-p\cdot y^{\hat{h}}\right]}, \end{aligned}$$
and use the shorthand \({u_s^h}^{\prime }:={u^h}^{\prime }\!\left[\omega ^h_s+r_s\cdot y^h\right]\) and \({u_s^h}^{\prime \prime }:={u^h}^{\prime \prime }\!\left[\omega ^h_s+r_s\cdot y^h\right]\). Then, (5) can be written as
$$\begin{aligned} \sum _{s\in S}\overline{\pi }_{s} \left({u^h_s}^{\prime }- \mu ^{h\hat{h}}\,{u^{\hat{h}}_s}^{\prime }\right)r_s =0, \quad \forall h,\hat{h}\in H. \end{aligned}$$
(6)
Since the subspaces \(L_1,\ldots ,L_K\) are linearly independent, the following result is immediate.

Theorem 3.1

At any equilibrium \((y,p)\) of \(\omega \in \varOmega \),
$$\begin{aligned} \sum _{s\in S_k}\overline{\pi }_s \left({u^h_s}^{\prime }- \mu ^{h\hat{h}}\,{u^{\hat{h}}_s}^{\prime }\right)r_s =0, \quad \forall h,\hat{h}\in H; \, S_k\in \mathcal S (R). \end{aligned}$$

In other words, at a competitive equilibrium, asset valuations are equalized across agents not only unconditionally, but also conditional on any equal-valuation event (or union of equal-valuation events). It follows that if, after trading, agents were to receive a public signal that induces the equal-valuation partition, they would not retrade since their asset valuations are already equal conditional on this partition.

Specializing Theorem 3.1 to an insurable state, we have the standard result:

Corollary 3.1

If \(s\) is an insurable state, then at any equilibrium \((y,p)\) of \(\omega \in \varOmega \), \({u^h_s}^{\prime }- \mu ^{h\hat{h}}\,{u^{\hat{h}}_s}^{\prime }=0\), for all \(h,\hat{h}\in H\).

For a generic subset of endowments, the converse of Theorem 3.1 is true as well, so that we can strengthen the result as follows.

Theorem 3.2

There is a generic subset \(\hat{\varOmega }\) of \(\,\varOmega \), such that at any equilibrium \((y,p)\) of \(\omega \in \hat{\varOmega }\),
$$\begin{aligned} \sum _{s\in \hat{S}}\overline{\pi }_s\left({u^h_s}^{\prime }- \mu ^{h\hat{h}}\,{u^{\hat{h}}_s}^{\prime }\right)r_s=0, \quad \forall h,\hat{h}\in H, \end{aligned}$$
(7)
if and only if \(\hat{S}\) is a union of equal-valuation events.

Thus, for a generic subset of endowments, the equal-valuation partition \(\mathcal S (R)\) is the finest partition of \(S\) conditional on which agents’ asset valuations are equal in equilibrium.

The proof of Theorem 3.2 uses the transversality theorem. Since we also exploit transversality in the proofs of Theorems 4.1 and 4.2 in the next section, it is useful to summarize the argument here. Consider a function \({\varPsi :\mathbb R ^n\times \mathcal E \rightarrow \mathbb R ^m}\), where \(\mathcal E \) is an open subset of Euclidean space and \(m>n\). For \(e\in \mathcal E \), let \(\varPsi _e\) be the function \(\varPsi (\cdot ,e)\). The argument involves identifying such a function \(\varPsi \), such that the desired result can be formulated as \(\varPsi _{e}^{-1}(0)=\varnothing \), for every \(e\) in a generic subset of \(\mathcal E \). We show that the Jacobian \(D_{x,e}\varPsi \) has full row rank at all zeros \((x,e)\) of \(\varPsi \), that is, \(\varPsi \) is transverse to zero. By the transversality theorem, there is then a dense subset \(\hat{\mathcal{E }}\) of \(\mathcal E \) such that, for each \(e\in \hat{\mathcal{E }}\), \(\varPsi _{e}:\mathbb R ^n\rightarrow \mathbb R ^m\) is transverse to zero. It follows that \(\varPsi _{e}^{-1}(0)=\varnothing \). In other words, the equation system \(\varPsi _e(x)=0\) has no solution since the number of (locally) independent equations \(m\) exceeds the number of unknowns \(n\). A standard argument (see, for example, Citanna et al. 1998) establishes that the set \(\hat{\mathcal{E }}\) is open and hence a generic subset of \(\mathcal E \).

Proof of Theorem 3.2

We begin by characterizing a competitive equilibrium as a solution to a system of (locally) independent equations. Let \(g(y)=0\) and \(f(y,p, \omega )=0\) denote the equation systems (2) and (4), respectively. The tuple \((y,p)\) is a competitive equilibrium for economy \(\omega \) if and only if it satisfies
$$\begin{aligned} F(y,p,\omega ) := \left( \begin{array}{c} f(y,p, \omega ) \\ g(y) \end{array} \right) = 0, \end{aligned}$$
(8)
which consists of \(JH+J\) equations, equal to the number of unknowns \((y,p)\). The Jacobian of \(F\) can be written as follows:
$$\begin{aligned} D_{y,p,\omega }F=\left( \begin{array}{cc} D_{y,p}f&D_{\omega }f \\ D_{y,p}g&0 \end{array} \right)_{,} \end{aligned}$$
with
$$\begin{aligned} D_{\omega }f=\mathrm{diag}_h\left[\ldots \overline{\pi }_s \, {u^h_s}^{\prime \prime } r_{s} \ldots _s\right] \end{aligned}$$
and
$$\begin{aligned} D_{y,p}g=\left( \ldots I_J \ldots _{h} \quad 0 \right), \end{aligned}$$
where \(I_J\) is the \(J\times J\) identity matrix. The matrix \(D_{\omega }f\) has full row rank since \(R\) has full column rank. Clearly, \(D_{y,p}g\) has full row rank as well.

We now proceed with the proof of the theorem. If \(\hat{S}\) is a union of equal-valuation events, Eq. (7) holds due to Theorem 3.1. To prove the converse, suppose \(\hat{S}\) is not a union of equal-valuation events. Then, there is a nontrivial equal-valuation event, which we can take to be \(S_1\) without loss of generality, such that \(\hat{S}\) contains some but not all elements of \(S_1\). Hence, we can write \(S_1\) as the union of two nonempty and disjoint sets, \(\hat{S_1}:=S_1\cap \hat{S}\) and \(\check{S}_1:= S_1 \backslash \hat{S}_1\). We reorder the set \(S_1\) so that the states in \(\hat{S}_1\) appear before the states in \(\check{S}_1\).

Recall that \(R_1\) is the first diagonal block of \(R^{\star }\) corresponding to the equal-valuation event \(S_1\). \(R_1\) can be partitioned as follows:
$$\begin{aligned} \left( \frac{\hat{R}_1}{\check{R}_1} \right), \end{aligned}$$
where \(\hat{R}_1\) consists of the rows of \(R_1\) corresponding to the states in \(\hat{S}_1\), while \(\check{R}_1\) consists of the remaining rows of \(R_1\), that is, those corresponding to the states in \(\check{S}_1\). Consider the \(S_1\times 2J_1\) matrix
https://static-content.springer.com/image/art%3A10.1007%2Fs00199-012-0717-z/MediaObjects/199_2012_717_Equa11_HTML.gif
This matrix is column-equivalent to a block-diagonal matrix whose diagonal blocks are \(\hat{R}_1\) and \(\check{R}_1\). It follows that \(\mathrm{rank}(\hat{Q}_1) = \mathrm{rank}(\hat{R}_1)+\mathrm{rank}(\check{R}_1)\). Since \(S_1\) is an equal-valuation event, the row spaces of \(\hat{R}_1\) and \(\check{R}_1\) have a nontrivial intersection, implying that \(\mathrm{rank}(\hat{R}_1)+\mathrm{rank}(\check{R}_1) > \mathrm{rank}(R_1)=J_1\). It follows that \(\mathrm{rank}(\hat{Q}_1)> J_1\).
Let \(\hat{r}^j:=[\ldots r^j_s \ldots _{s\in \hat{S}_1} \quad 0_{1\times \check{S}_1}]^{\top }\), a vector in \(\mathbb R ^{S_1}\). Since \(\mathrm{rank}(\hat{Q}_1)> J_1\), we can pick \(j\in J_1\) such that \(\hat{r}^j\) lies outside the column span of \(R_1\) (in other words, we can choose one of the first \(J_1\) columns of \(\hat{Q}_1\) such that it is outside the span of the last \(J_1\) columns of \(\hat{Q}_1\)). We fix such a value of \(j\) for the remainder of the proof. Due to the block structure of \(R\) given by (3), the vector \([\hat{r}^j \quad 0_{1\times (S-S_1)}]^{\top }=[\ldots r^j_s \ldots _{s\in \hat{S}_1} \quad 0_{1\times (S-\hat{S}_1)}]^{\top }\) lies outside the column span of \(R\). In other words, the matrix
$$\begin{aligned} A:= \left( \frac{\ldots r^j_s \ldots _{s\in \hat{S}_1} \qquad 0_{1\times (S-\hat{S}_1)}}{\ldots r_s \ldots _{s\in S}} \right) \end{aligned}$$
(9)
has full row rank \(J+1\). We will use this fact below.
It suffices to establish the theorem for the first two agents, \(h_1\) and \(h_2\). We will show that, for \(\omega \) in a generic subset of \(\varOmega \), there is no solution to the equation system
$$\begin{aligned} \varPsi _1(y,p,\omega ):=\left( \begin{array}{c} F(y,p,\omega ) \\ \sum _{s\in \hat{S}}\overline{\pi }_s\left({u^{h_1}_s}^{\!\prime }- \mu ^{h_1h_2}\,{u^{h_2}_s}^{\!\prime }\right)r^j_s \end{array} \right) =0. \end{aligned}$$
Since \(j\in J_1\), \(r^j_s=0\) for all \(s\notin S_1\), so that the sum over \(\hat{S}\) in this equation system can be restricted to \(\hat{S}_1\). Hence, the Jacobian \(D_{y,p,\omega }\varPsi _1\), evaluated at a zero \((y,p,\omega )\) of \(\varPsi _1\), is
https://static-content.springer.com/image/art%3A10.1007%2Fs00199-012-0717-z/MediaObjects/199_2012_717_Equa13_HTML.gif
The Jacobian is row-equivalent to
https://static-content.springer.com/image/art%3A10.1007%2Fs00199-012-0717-z/MediaObjects/199_2012_717_Equa14_HTML.gif
which in turn is column-equivalent to
https://static-content.springer.com/image/art%3A10.1007%2Fs00199-012-0717-z/MediaObjects/199_2012_717_Equa15_HTML.gif
where \(A\) is defined by (9). This matrix has full row rank since each of the diagonal blocks has that property. Therefore, so does the Jacobian \(D_{y,p,\omega }\varPsi _1\), at every zero of \(\varPsi _1\). Thus, \(\varPsi _1\) is transverse to zero, and \(\varPsi _{1\omega }^{-1}(0)=\varnothing \) for all \(\omega \) in a generic subset of \(\varOmega \). \(\square \)

The following result is an immediate consequence of Theorem 3.2:

Corollary 3.2

A state \(s\in S^{\star }\) is an insurable state if and only if, at any equilibrium \((y,p)\) of \(\omega \in \hat{\varOmega }\), \({u^h_s}^{\prime }- \mu ^{h\hat{h}}\,{u^{\hat{h}}_s}^{\prime }=0\), for all \(h,\hat{h}\in H\).

This result can be established directly for a generic subset of endowments using standard arguments.

4 Information and retrading

We wish to describe the kinds of (unanticipated) information that will induce agents to retrade. We assume that the information takes the form of a public signal correlated with the state of the world \(s\) that agents observe after trading at date 0, but before consumption takes place,8 and before the uncertainty regarding endowments and asset payoffs is resolved. We consider the class of public signals that take finitely many values. Accordingly, we fix a finite set of possible “signal realizations” \(\varSigma \), \(\#\varSigma \ge 2\), with a typical element of \(\varSigma \) denoted by \(\sigma \). A public signal can then be described by a probability measure on \(S\times \varSigma \), that is, by the probabilities \(\pi :=\left\{ \pi _{s\sigma }\right\} _{s\in S,\sigma \in \varSigma }\in \mathbb R _{+}^{S\varSigma }\), where \(\pi _{s\sigma }\) denotes \(\mathrm{Prob}(s,\sigma )\). Let \(\pi _{s} := \mathrm{Prob}(s)=\sum _{\sigma }\pi _{s\sigma }, \, \pi _{\sigma }:=\mathrm{Prob}(\sigma )=\sum _s\pi _{s\sigma }\), and \(\pi _{s|\sigma }:=\mathrm{Prob}(s|\sigma )=\pi _{s\sigma }/\pi _{\sigma }\).

Since a public signal is completely described by the associated vector \(\pi \), we refer to \(\pi \) itself as a public signal. Formally, a public signal lies in the set
$$\begin{aligned} \varPi := \left\{ \pi \in \mathbb R ^{S\varSigma }_+ \,\big \vert \, \pi _s=\overline{\pi }_s, \forall s\in S; \, \pi _{\sigma }>0, \forall \sigma \in \varSigma \right\} . \end{aligned}$$
In other words, any public signal in \(\varPi \) is consistent with the uncertainty over fundamentals given by \(\{\overline{\pi }_s\}_{s\in S}\). The condition on the marginal distribution over \(\varSigma \) is without loss of generality. This specification admits a range of possible signals. It includes those that have full support, with \({\{s\in S \, \vert \, \pi _{s|\sigma }>0\}}=S\), for all \(\sigma \). It also includes signals for which the support \({\{s\in S \, \vert \, \pi _{s|\sigma }>0\}}\) is a strict subset of \(S\) for some signal realizations. A special case of the latter is one where the signal induces a partition of \(S\).9
In the remainder of this section, we characterize the set of public signals that lead to retrade. Given an equilibrium \((y,p)\), there is no retrade at \(\pi \) if and only if the equality of asset valuations which holds in equilibrium (condition (6)) also holds at \(\pi \), that is,
$$\begin{aligned} \sum _{s\in S}\pi _{s|\sigma } \left({u^h_s}^{\prime }- \mu ^{h\hat{h}}\,{u^{\hat{h}}_s}^{\prime }\right)r_s =0, \quad \forall h,\hat{h}\in H; \, \sigma \in \varSigma . \end{aligned}$$
As in Theorem 3.1, we can exploit the linear independence of the subspaces \(L_1,\ldots ,L_K\) to refine this no-retrade condition.

Lemma 4.1

Given an equilibrium \((y,p)\), there is no retrade at \(\pi \) if and only if
$$\begin{aligned} \sum _{s\in S_k}\pi _{s|\sigma } \left({u^h_s}^{\prime }- \mu ^{h\hat{h}}\,{u^{\hat{h}}_s}^{\prime }\right)r_s =0, \quad \forall h,\hat{h}\in H; \, S_k\in \mathcal S (R); \, \sigma \in \varSigma . \end{aligned}$$
(10)
It is clear from (10) that a public signal that affects only the relative likelihood of equal-valuation events does not generate retrade. Agents’ asset valuations remain equal after the arrival of such a signal. For example, a public signal that induces a partition of \(S\) that contains the equal-valuation partition \(\mathcal S (R)\), or is equal to \(\mathcal S (R)\), does not generate any retrade since it leaves the conditional distribution over \(S_k\) unchanged for every \(k\). More generally, if \(\pi \) leads to retrade, it must belong to the following set:
$$\begin{aligned} \hat{\varPi } := \left\{ \pi \in \varPi \,\big \vert \, \exists \, \sigma \in \varSigma , \, S_k\in \mathcal S (R) \,\, \text{ s.t.} \,\, \{\pi _{s|\sigma }\}_{s\in S_k} \ne \alpha \{\overline{\pi }_s\}_{s\in S_k}, \forall \alpha \ge 0 \right\} . \end{aligned}$$
This is the set of public signals \(\pi \) for which \(\{\pi _{s|\sigma }\}_{s\in S_k}\) is not proportional to \(\{\overline{\pi }_s\}_{s\in S_k}\) for some \(\sigma \) and some equal-valuation event \(S_k\). Of course, \(S_k\) must be nontrivial for this to be the case. Thus, \(\hat{\varPi }\) is empty if markets are \(S^{\star }\)-complete. On the other hand, if markets are \(S^{\star }\)-incomplete, \(\hat{\varPi }\) is a generic subset of \(\varPi \).10 It includes both partitional and nonpartitional information structures. In fact, among signals that induce a partition of \(S\), the only ones excluded from \(\hat{\varPi }\) are those for which the partition is (weakly) coarser than \(\mathcal S (R)\).

If markets are \(S^{\star }\)-incomplete, for the generic subset of endowments \(\hat{\varOmega }\) for which Theorem 3.2 (and hence Corollary 3.2) holds, \({u^h_s}^{\prime }- \mu ^{h\hat{h}}\,{u^{\hat{h}}_s}^{\prime }\ne 0\) for every state \(s\) in a nontrivial equal-valuation event \(S_k\). While the no-retrade condition (10) is not necessarily violated for every \(\pi \in \hat{\varPi }\) and \(\omega \in \hat{\varOmega }\), we show that an appropriate perturbation of either \(\pi \) or \(\omega \) ensures that it is violated. More precisely, Theorem 4.1 establishes that, for every \(\pi \in \hat{\varPi }\), retrade occurs for \(\omega \) in a generic subset of \(\hat{\varOmega }\) (and hence of \(\varOmega \)).11 Analogously, Theorem 4.2 shows that, for every \(\omega \in \hat{\varOmega }\), retrade occurs for \(\pi \) in a generic subset of \(\hat{\varPi }\) (and hence of \(\varPi \)). Finally, Theorem 4.3 strengthens Theorem 4.2 by showing that retrade occurs for every public signal that is “sufficiently rich”, in a sense that we shall make precise.

We say that an economy \(\omega \)admits a  \(\pi \)-retrade if at every equilibrium of this economy the public signal \(\pi \) leads to retrade for at least one value of \(\sigma \).

Theorem 4.1

Suppose markets are \(S^{\star }\)-incomplete. Then, for any \(\pi \in \hat{\varPi }\), there is a generic subset \(\check{\varOmega }(\pi )\) of \(\,\hat{\varOmega }\) such that every economy \(\omega \in \check{\varOmega }(\pi )\) admits a \(\pi \)-retrade.

Thus, for \(\pi \) to lead to retrade, it is not only necessary that it belong to \(\hat{\varPi }\) but, for a generic subset of endowments, sufficient as well. As noted above, \(\hat{\varPi }\) is a generic subset of \(\varPi \) that contains both partitional and nonpartitional information. An example of partitional information that leads to retrade (in fact, for any \(\omega \in \hat{\varOmega }\)) is provided in Example 3 at the end of this section.

Proof of Theorem 4.1

Consider a \(\pi \) in \(\hat{\varPi }\) and fix a \(\sigma \) and a nontrivial equal-valuation event, which we can take to be \(S_1\) without loss of generality, such that \(\{\pi _{s|\sigma }\}_{s\in S_1}\) is not proportional to \(\{\overline{\pi }_s\}_{s\in S_1}\). Let
$$\begin{aligned} Q_1:= \left( \mathrm{diag}_{s\in S_1}[\pi _{s|\sigma }]R_1 \quad \mathrm{diag}_{s\in S_1}[\overline{\pi }_s] R_1 \right). \end{aligned}$$
We claim that \(\mathrm{rank}(Q_1)> J_1\). If \(\pi _{s|\sigma }>0\) for all \(s\in S_1\), this is immediate from the following result, which can be deduced from Lemma 5 of Geanakoplos and Mas-Colell (1989):

Fact 1 Let \(R_k\) be the diagonal block of \(R^{\star }\) corresponding to a nontrivial equal-valuation event \(S_k\in \mathcal S (R)\).  Consider nonzero scalars \(\theta _s,\theta ^{\prime }_s\)\(s\in S_k\)such that  \(\{\theta _s\}_{s\in S_k}\)  is not proportional to  \(\{\theta ^{\prime }_s\}_{s\in S_k}\)Then\(\mathrm{diag}_{s\in S_k}[\theta _s] R_k\) and \(\mathrm{diag}_{s\in S_k}[\theta ^{\prime }_s] R_k\) do not have the same column span.

Suppose, on the other hand, that \(\pi _{s|\sigma }=0\) for some \(s\in S_1\). Let \(\mathring{S}_1\) be the set of states in \(S_1\) for which \(\pi _{s|\sigma }=0\), and let \(\mathring{R}_1\) be the \(\mathring{S}_1 \times J_1\) submatrix of \(R_1\) corresponding to these states. Similarly, let \(\breve{S}_1\) be the remaining states in \(S_1\) and \(\breve{R}_1\) the submatrix of \(R_1\) corresponding to these states. Then, \(Q_1\) is row-equivalent to
https://static-content.springer.com/image/art%3A10.1007%2Fs00199-012-0717-z/MediaObjects/199_2012_717_Equa20_HTML.gif
If we delete the rows of \(\breve{Q}_1\) corresponding to the redundant rows of its upper left block, and also delete the rows corresponding to the redundant rows of its lower right block, we are left with a block-triangular matrix whose diagonal blocks have full row rank and hence whose rank is equal to the sum of the ranks of the diagonal blocks. It follows that for the full matrix \(\breve{Q}_1\), \(\mathrm{rank}(\breve{Q}_1)\ge \mathrm{rank}(\breve{R}_1)+\mathrm{rank}(\mathring{R}_1)\). Since \(S_1\) is an insurable event, the row spaces of \(\breve{R}_1\) and \(\mathring{R}_1\) have a nontrivial intersection, implying that \(\mathrm{rank}(\breve{R}_1)+\mathrm{rank}(\mathring{R}_1)>\mathrm{rank}(R_1)=J_1\). This in turn implies that the rank of \(\breve{Q}_1\), which is equal to the rank of \(Q_1\), is strictly greater than \(J_1\).
Consequently, the rank of
$$\begin{aligned} Q:= \left( \mathrm{diag}_{s\in S}[\pi _{s|\sigma }]R \quad \mathrm{diag}_{s\in S}[\overline{\pi }_s] R \right) \end{aligned}$$
is strictly greater than \(J\). Therefore, we can pick \(j\) such that \(\mathrm{diag}_{s\in S}[\pi _{s|\sigma }]\,r^j\) lies outside the column span of \(\mathrm{diag}_{s\in S}[\overline{\pi }_s] R\), so that the matrix
$$\begin{aligned} B:=\left( \begin{array}{c} \ldots \pi _{s|\sigma }\, r^j_s \ldots _{s\in S} \\ \ldots \overline{\pi }_s\, r_s \ldots _{s\in S} \end{array} \right) \end{aligned}$$
(11)
has full row rank \(J+1\). We fix such a value of \(j\) for the remainder of the proof.
Recall that the equations describing an equilibrium are given by \(F(y,p,\omega )=0\) (Eq. (8)). We will show that, for a generic subset of \(\varOmega \), there is no solution to the equation system
$$\begin{aligned} \varPsi _2(y,p,\omega ;\pi ):=\left( \begin{array}{c} F(y,p,\omega ) \\ \sum _{s\in S}\pi _{s|\sigma }\left( {u_s^{h_1}}^{\!\prime } - \mu ^{h_1h_2}{u_s^{h_2}}^{\prime } \right)r^j_s \end{array} \right) =0, \end{aligned}$$
that is, the no-retrade condition (10) is violated for the first two agents, \(h_1\) and \(h_2\). The Jacobian of \(\varPsi _2\) is
https://static-content.springer.com/image/art%3A10.1007%2Fs00199-012-0717-z/MediaObjects/199_2012_717_Equa23_HTML.gif
The Jacobian is row-equivalent to
https://static-content.springer.com/image/art%3A10.1007%2Fs00199-012-0717-z/MediaObjects/199_2012_717_Equa24_HTML.gif
which in turn is column-equivalent to
https://static-content.springer.com/image/art%3A10.1007%2Fs00199-012-0717-z/MediaObjects/199_2012_717_Equa25_HTML.gif
where \(B\) is defined by (11). This matrix has full row rank since each of the diagonal blocks has that property. Therefore, so does the Jacobian \(D_{y,p,\omega }\varPsi _2\), at every zero of \(\varPsi _2\). Thus, \(\varPsi _2\) is transverse to zero, and \(\varPsi _{2\omega }^{-1}(0)=\varnothing \), for every \(\omega \) in a generic subset of \(\varOmega \). This generic subset depends on \(\pi \), which is a parameter of \(\varPsi _2\). Moreover, by taking the intersection of this set with \(\hat{\varOmega }\), we obtain the generic subset \(\check{\varOmega }(\pi )\).12\(\square \)

Next, we present our second retrading result which involves perturbing probabilities.

Theorem 4.2

Suppose markets are \(S^{\star }\)-incomplete. Then, every economy \(\omega \in \hat{\varOmega }\) admits a \(\pi \)-retrade for every \(\pi \in \hat{\varPi }_1(\omega )\), a generic subset of \(\hat{\varPi }\).

Proof

Fix a pair of agents \(h\) and \(\hat{h}\), a nontrivial equal-valuation event \(S_k\), and a signal realization \(\sigma \). It suffices to show that the no-retrade condition (10) is violated for these given values. Let \(\hat{s}\) be a state in \(S_k\), and let \(j\in J_k\) be an asset which has a nonzero payoff in \(\hat{s}\), that is, \(r^j_{\hat{s}}\ne 0\). Such an asset exists since no state in \(S_k\) is a zero-payoff state.

Consider an economy \(\omega \in \hat{\varOmega }\). We will show that, for \(\pi _{\hat{s}\sigma }\) in a generic subset of the interval \((0,\overline{\pi }_{\hat{s}})\), at every equilibrium \((y,p)\), there is no solution to the equation system
$$\begin{aligned} \varPsi _3(y,p,\pi _{\hat{s}\sigma };\omega ):=\left( \begin{array}{c} F(y,p;\omega ) \\ \sum _{s\in S_k}\pi _{s\sigma } \left({u^h_s}^{\prime }- \mu ^{h\hat{h}}\,{u^{\hat{h}}_s}^{\prime }\right)r^j_s \end{array} \right) =0, \end{aligned}$$
and thus, the no-retrade condition (10) is violated. For any choice of \(\pi _{\hat{s}\sigma }\in (0,\overline{\pi }_{\hat{s}})\), we can always choose \(\{\pi _{\hat{s}\sigma ^{\prime }}\}_{\sigma ^{\prime }\ne \sigma }\), so that \(\sum _{\sigma }\pi _{\hat{s}\sigma }=\overline{\pi }_{\hat{s}}\). Moreover, if \(\pi _{\hat{s}\sigma }\) is in a generic subset of \((0,\overline{\pi }_{\hat{s}})\), a corresponding \(\pi \) is in a generic subset of \(\varPi \). Clearly, \(\pi \) must also lie in \(\hat{\varPi }\) and hence belongs to a generic subset of \(\hat{\varPi }\).
The Jacobian of \(\varPsi _3\), evaluated at a zero \((y,p, \pi _{\hat{s}\sigma })\) of \(\varPsi _3\), is
https://static-content.springer.com/image/art%3A10.1007%2Fs00199-012-0717-z/MediaObjects/199_2012_717_Equa12_HTML.gif
(12)
Since \(\hat{s}\) is not an insurable state, \({u^h_{\hat{s}}}^{\prime }- \mu ^{h\hat{h}}\,{u^{\hat{h}}_{\hat{s}}}^{\prime }\ne 0\), by Corollary 3.2. Also, we have chosen asset \(j\) for which \(r^j_{\hat{s}}\) is nonzero. Hence, the lower right block of (12) is a nonzero scalar. Moreover, for \(\omega \in \hat{\varOmega }\), we see from the proof of Theorem 3.2 that \(D_{y,p}\varPsi _1\) has full rank and, therefore, so does \(D_{y,p}F\), at all zeros of \(F\).

Therefore, the Jacobian \(D_{y,p,\pi _{\hat{s}\sigma }}\varPsi _3\) has full row rank, at every zero of \(\varPsi _3\). Thus, \(\varPsi _3\) is transverse to zero, and \(\varPsi _{3\pi _{\hat{s}\sigma }}^{-1}(0)=\varnothing \), for every \(\pi _{\hat{s}\sigma }\) in a generic subset of \((0,\overline{\pi }_{\hat{s}})\). This generic subset depends on \(\omega \), which is a parameter of \(\varPsi _3\). \(\square \)

In Theorem 4.1, the no-retrade condition (10) is violated by fixing a \(\pi \) in \(\hat{\varPi }\) and perturbing endowments. The generic set \(\check{\varOmega }(\pi )\), therefore, depends on \(\pi \). In Theorem 4.2, on the other hand, a violation of the no-retrade condition is achieved by fixing an \(\omega \) in \(\hat{\varOmega }\) and perturbing \(\pi \). The generic set \(\hat{\varPi }_1(\omega )\), therefore, depends upon \(\omega \). In our final result, we consider economies in \(\hat{\varOmega }\), as in Theorem 4.2, and identify a subset of \(\hat{\varPi }\) of “sufficiently rich” public signals, which does not depend on the economy under consideration, such that retrade occurs for every \(\pi \) in this set. A signal is “sufficiently rich” if it changes the relative probabilities of states in some nontrivial equal-valuation event \(S_k\) not just for one value of \(\sigma \) (as is the case for every \(\pi \in \hat{\varPi }\)), but independently for a number of values of \(\sigma \) that exceeds the degree of market incompleteness, \(S_k-J_k\), in the event \(S_k\). More precisely, we establish the result for the following set of public signals:
$$\begin{aligned} \hat{\varPi }_2 := \left\{ \pi \in \varPi \,\big \vert \, \exists \, S_k\in \mathcal S (R) \,\, \text{ s.t.} \,\, \mathrm{rank}(\varLambda _{\pi ,S_k})>S_k-J_k>0 \right\} , \end{aligned}$$
where
$$\begin{aligned} \varLambda _{\pi ,S_k}:= \left( \begin{array}{l} \qquad \vdots \\ \ldots \pi _{s|\sigma } \ldots _{s\in S_k} \\ \qquad \vdots _{\sigma } \end{array} \right). \end{aligned}$$
(13)
Clearly, \(\hat{\varPi }_2\) is a generic subset of \(\hat{\varPi }\).13

Theorem 4.3

Suppose markets are \(S^{\star }\)-incomplete. Then, every economy \(\omega \in \hat{\varOmega }\) admits a \(\pi \)-retrade, for every \(\pi \in \hat{\varPi }_2\).

Proof

Consider an economy \(\omega \in \hat{\varOmega }\), an equilibrium \((y,p)\), a nontrivial equal-valuation event \(S_k\), and a \(\pi \in \hat{\varPi }\) satisfying \(\mathrm{rank}(\varLambda _{\pi ,S_k})>S_k-J_k\). Suppose there is no retrade at \(\pi \). Then, the no-retrade condition (10) holds for event \(S_k\), for an arbitrary pair of agents \(h\) and \(\hat{h}\):
$$\begin{aligned} \sum _{s\in S_k}\pi _{s|\sigma }\left({u^h_s}^{\prime }- \mu ^{h\hat{h}}\,{u^{\hat{h}}_s}^{\prime }\right)r_s =0, \quad \forall \sigma \in \varSigma , \end{aligned}$$
which can be rewritten as follows:
$$\begin{aligned} \varLambda _{\pi ,S_k}\,\mathrm{diag}_{s\in S_k}\!\left[\left({u^h_s}^{\prime }- \mu ^{h\hat{h}}\,{u^{\hat{h}}_s}^{\prime }\right)\right]R_k =0. \end{aligned}$$
(14)
By Corollary 3.2, \({u^h_s}^{\prime }- \mu ^{h\hat{h}}\,{u^{\hat{h}}_s}^{\prime }\ne 0,\) for all \(s\in S_k\). Therefore, the rank of \(D_k:=\mathrm{diag}_{s\in S_k}\![({u^h_s}^{\prime }- \mu ^{h\hat{h}}\,{u^{\hat{h}}_s}^{\prime })]R_k\) is \(J_k\). Let \(\mathcal D _k\) be the column space of \(D_k\). Equation (14) implies that the rows of \(\varLambda _{\pi ,S_k}\) lie in \(\mathcal D _k^{\perp }\), the orthogonal complement of \(\mathcal D _k\) in \(\mathbb R ^{S_k}\). It follows that \(\mathrm{rank}(\varLambda _{\pi ,S_k})\le \mathrm{dim}(\mathcal D _k^{\perp })=S_k-J_k\), a contradiction. \(\square \)

The theorem generalizes Theorem 5 of Blume et al. (2006). They impose a stronger full rank condition on the public signal; in our notation, their assumption is that \({\mathrm{rank}(\varLambda _{\pi ,S_1} \ldots \varLambda _{\pi ,S_K})=S}\) or that the matrix (13) defined over \(S\) rather than \(S_k\) has full column rank. Moreover, they only consider public signals that have full support and hence do not include those that induce a partition of \(S\). They also assume that there is an insurable state and that for each state there is at least one asset whose payoff is positive in that state.

While in Theorems 4.1 and 4.2 it sufficed to consider a public signal for only two values of \(\sigma \), for example, an appropriate choice of \(\{\pi _{s|\sigma _1}\}_{s\in S_k}\) for which there is retrade conditional on \(\sigma _1\), and a corresponding choice of \(\{\pi _{s|\sigma _2}\}_{s\in S_k}\), \(\pi _{\sigma _1}\) and \(\pi _{\sigma _2}\) in order to ensure that \(\pi _{s\sigma _1}+\pi _{s\sigma _2}=\overline{\pi }_s\) for all \(s\in S_k\), Theorem 4.3 requires an independent change in information across at least \(S_k-J_k\) values of \(\sigma \).

The sets \(\hat{\varPi }_1(\omega )\) and \(\hat{\varPi }_2\), that is, the generic subsets of \(\hat{\varPi }\) identified by Theorems 4.2 and 4.3 for which retrade occurs, are not nested in general.

Example 3

Suppose there are three equally likely states, that is, \(S=\{s_1,s_2,s_3\}\) and \(\overline{\pi }_s=1/3\) for all \(s\), and the asset payoff matrix is given by
$$\begin{aligned} R= \left( \begin{array}{cc} 1&\quad 0 \\ 1&\quad 0 \\ 0&\quad 1 \end{array} \right). \end{aligned}$$
Then, the equal-valuation partition \(\mathcal S (R)\) is given by \(\{S_1,S_2\}\), where \(S_1=\{s_1,s_2\}\) is a nontrivial equal-valuation event and \(S_2\) consists of the single insurable state \(s_3\). Suppose \(\varSigma =\{\sigma _1,\sigma _2\}\). Consider a public signal that induces the partition \(\{S_1,S_2\}\), that is, the conditional probabilities over the three states are given by \((1/2,1/2,0)\) for one value of \(\sigma \) and \((0,0,1)\) for the other. This signal is not in \(\hat{\varPi }\) and, therefore, does not generate any retrade. On the other hand, consider a signal that induces the partition \(\{\{s_1\},\{s_2,s_3\}\}\), for example, with the conditional probabilities given by \((1,0,0)\) for \(\sigma _1\) and \((0,1/2,1/2)\) for \(\sigma _2\). This signal does lie in \(\hat{\varPi }\). In fact, it lies in \(\hat{\varPi }_2\) since
$$\begin{aligned} \varLambda _{\pi ,S_1} = \left( \begin{array}{cc} 1&\quad 0 \\ 0&\quad \frac{1}{2} \end{array} \right), \end{aligned}$$
which has rank equal to 2, greater than \(S_1-J_1=1\). By Theorem 4.3, there is retrade for every economy in \(\hat{\varOmega }\). Of course, it is not necessary that the public signal induce a partition of \(S\) in order to generate retrade. \(\quad \Vert \)
Footnotes
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 \(\sigma \).

 

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