Participation in risk sharing under ambiguity

Abstract

This paper is about (non) participation in efficient risk sharing among agents who have ambiguous beliefs about uncertain states of nature. The question we ask is whether and how can ambiguous beliefs give rise to some agents not participating in efficient risk sharing. Ambiguity of beliefs is described by the multiple-prior expected utility of Gilboa and Schmeidler (J Math Econ 18:141–153, 1989), or the variational preferences of Maccheroni et al. (Econometrica 74(6):1447–1498, 2006). The main result says that if the aggregate risk is relatively small, then the agents whose beliefs are the most ambiguous do not participate in risk sharing. The higher the ambiguity of those agents’ beliefs, the more likely is their non-participation. Another factor making non-participation more likely is low risk aversion of agents whose beliefs are less ambiguous. We discuss implications of our results on agents’ participation in trade in equilibrium in assets markets.

Appendix

For two probability measures $P, Q \in \Delta ,$ let $|P - Q|$ denote the total-variation distance between them. That is,

$$\begin{aligned} |P - Q| = \sum _{s=1}^S |P(s) - Q(s)|. \end{aligned}$$

(11)

Further, let $V_{\delta } (\mathcal{P}) \subset \Delta$ denote the $\delta$-neighborhood of the set $\mathcal{P} \subset \Delta$ in the variational distance for $\delta > 0.$ Let ${\hat{\omega }}= \max _s \omega _s$ and let $\hat{A}_i = \sup \{A_i(x) : x \in [0, {\hat{\omega }}]\}$ where $A_i(x) = - {v_i''(x) \over v_i'(x)}$ is the Arrow-Pratt measure of risk aversion.

Proof of Theorem 1

First we show the following lemma.

Lemma 2

If $c \in B_{\epsilon }(D)$ and $c \in R^S_{++},$ then $Q_i(c) \subset V_{\delta } (\mathcal{P}_i)$ for $\delta = e^{2 \epsilon \hat{A}_i} - 1.$

Proof

Take any $Q \in Q_i(c)$. Let $P\in \mathcal{P}_i$ be such that $Q(s) = {{v'_i(c(s)) P(s)} \over {\mathbf{E}_{P}[v'_i(c)]}}$ for every s. Further, let ${\underline{c}} = \min _s c_s$ and ${\bar{c}} = \max _s c_s.$ We have

$$\begin{aligned} |P - Q| = \sum _{s=1}^S P(s) {|\mathbf{E}_{P}[v'_i(c)] - v'_i(c(s))| \over {\mathbf{E}_{P}[v'_i(c)]}} \le {{v'_i({\underline{c}}) - v'_i(\bar{c})} \over v'_i({\bar{c}})} = {{v'_i({\underline{c}})} \over v'_i({\bar{c}})} - 1. \end{aligned}$$

(12)

Further,

$$\begin{aligned} \ln v'_i({\underline{c}}) - \ln v'_i(\bar{c}) = \int _{\underline{c}}^{\bar{c}} A_i(x) dx \le 2 \epsilon \hat{A}_i, \end{aligned}$$

(13)

where we used the fact that $\bar{c} - {\underline{c}} \le 2 \epsilon$ for $c \in B_{\epsilon }(D).$ Combining, we obtain

$$\begin{aligned} |P - Q| \le e^{2 \epsilon \hat{A}_i} - 1. \end{aligned}$$

(14)

Therefore $Q \in V_{\delta } (\mathcal{P}_i).$ $\square$

We proceed now with the proof of Theorem 1. Since $\mathcal{P}_j \subset \mathrm{int} \mathcal{P}_k,$ there exists $\delta$ be such that $V_{\delta } (\mathcal{P}_j) \subset \mathrm{int} \mathcal{P}_k.$ Let $\tilde{\epsilon }$ be such that $e^{2 \tilde{\epsilon }\hat{A}_j} - 1 = \delta .$ By Lemma 2, $Q_j(c_j) \subset \mathrm{int} \mathcal{P}_k$ for every $c_j \in B_{\tilde{\epsilon }}(D).$ If $c_j$ is part of an interior Pareto optimal allocation and $Q_j(c_j) \subset \mathrm{int} \mathcal{P}_k$, then it follows from Proposition 1 and Lemma 1 (ii) that $c_k$ is risk free. Hence, it suffices to show that there exists $\epsilon > 0$, such that if $\omega \in B_{\epsilon } (D),$ then $c_j \in B_{\tilde{\epsilon }}(D)$ for every consumption plan $c_j$ that is part of an interior Pareto optimal allocation.

Let $\mathcal{E}_j(\omega )$ be the set of Pareto optimal consumption plans of agent j. Let $b \in D$ be such that $b> > \omega .$ Mapping $\mathcal{E}_j(\cdot )$ is an upper hemi-continuous correspondence on the compact set [0, b]. Let $\hat{D} = D \cap [0, b].$ By assumption (8), $\mathcal{E}_j(\omega ) \subset \hat{D}$ if $\omega \in \hat{D}.$ Therefore there exists $\epsilon$ be such that if $\omega \in B_{\epsilon } (\hat{D}),$ then $\mathcal{E}_j(\omega ) \subset B_{\tilde{\epsilon }}(\hat{D}).$ This concludes the proof of Theorem 1. $\square$

Proof of Theorem 2

We start with a lemma.

Lemma 3

If $\hat{A}_i < \epsilon ,$ then $Q_i(c) \subset V_{\delta } (\mathcal{P}_i)$ for $\delta = e^{\epsilon {\hat{\omega }}} -1$ and every $c \in R^S_{++}$ such that $c \le \omega .$

Proof

The proof is similar to Lemma 2. For any $Q \in Q_i(c)$ let $P\in \mathcal{P}_i$ be such that $Q(s) = {{v'_i(c(s)) P(s)} \over {\mathbf{E}_{P}[v'_i(c)]}}$ for every s. We have

$$\begin{aligned} |P - Q| = \sum _{s=1}^S P(s) {|\mathbf{E}_{P}[v'_i(c)] - v'_i(c(s))| \over {\mathbf{E}_{P}[v'_i(c)]}} \le {{v'_i(0) - v'_i({\hat{\omega }})} \over v'_i({\hat{\omega }})}. \end{aligned}$$

(15)

Since

$$\begin{aligned} \ln v_i'(0) - \ln v_i'({\hat{\omega }}) = \int _0^{{\hat{\omega }}} A_i(x) dx \le \epsilon {\hat{\omega }}, \end{aligned}$$

(16)

it follows that

$$\begin{aligned} |P - Q| \le e^{\epsilon {\hat{\omega }}} - 1, \end{aligned}$$

(17)

Therefore $Q \in V_{\delta } (\mathcal{P}_i)$ for $\delta = e^{\epsilon {\hat{\omega }}} -1.$ $\square$

We proceed with the proof of Theorem 2. Since $\mathcal{P}_j \subset \mathrm{int} \mathcal{P}_k,$ there exists $\delta$ such that $V_{\delta } (\mathcal{P}_j) \subset \mathrm{int} \mathcal{P}_k.$ Let $\epsilon$ be such that $e^{\epsilon {\hat{\omega }}} - 1 = \delta .$ If $\hat{A}_j < \epsilon$ and $c_j$ is part of an interior Pareto optimal allocation $\{c_i\}$, then $Q_j(c_j) \subset \mathrm{int} \mathcal{P}_k,$ where we used Lemma 3. Using Proposition 1 and Lemma 1 (ii) we obtain that $c_k$ is risk free. $\square$

Proof of footnote 4 in Sect. 3:

Proposition 5

If $int \bigcap _{i=1}^I \mathcal{P}_i \ne \emptyset$ and there is no aggregate risk, then every Pareto optimal allocation is risk free.

Proof

Let $\{c_i\}$ be a feasible allocation. Let $\pi \in \mathrm{int} \bigcap _{i=1}^I \mathcal{P}_i.$ For each i,  let $\hat{c}_i = E_{\pi }(c_i).$ Since $\bar{\omega }$ is risk free, it follows that allocation $\{\hat{c}_i\}$ is feasible. Inequality (7) implies that $U_i(\hat{c}_i) \ge U_i(c_i),$ with strict inequality if $c_i$ is risky. It follows that every Pareto optimal allocation is risk free. $\square$