Assortative matching of risk-averse agents with endogenous risk

Abstract

A standard risk-sharing matching game predicts negative assortative matching over agents’ risk attitudes. In regards to risk sharing, less risk-averse agents prefer highly risk-averse partners, who pay a high risk premium. Negative sorting is, however, inconsistent with empirical and experimental literature. To resolve this conflict, we propose a model where agents can control the risks to their incomes. In regards to risk management, agents prefer similar partners because of their aligned objectives in risk management. When it is easy to control risks or all agents are sufficiently risk-averse, the risk-management effect dominates, leading to positive sorting.

Appendices

Appendix A: Proof of Lemma 1

Proof

F.O.C of (1) is

$$\begin{aligned} u_{i}^{\prime }( Y_{ij}-c_{ij}-\psi ( \mu _{i},\sigma _{i}^{2}) ) =\lambda u_{j}^{\prime }( c_{ij}-\psi ( \mu _{j},\sigma _{j}^{2}) ). \end{aligned}$$

Taking a log on both sides and use the specific form of CARA, we have

$$\begin{aligned} r_{i}( Y_{ij}-c_{ij}-\psi ( \mu _{i},\sigma _{i}^{2}) ) =r_{j}( c_{ij}-\psi ( \mu _{j},\sigma _{j}^{2}) ) -\log \lambda , \end{aligned}$$

which gives

$$\begin{aligned} c_{ij}=\frac{r_{i}Y_{ij}}{r_{i}+r_{j}}-\frac{r_{i}\psi ( \mu _{i} ,\sigma _{i}^{2}) -r_{j}\psi ( \mu _{j},\sigma _{j}^{2}) -\log \lambda }{r_{i}+r_{j}}, \end{aligned}$$

and

$$\begin{aligned} Y_{ij}-c_{ij}=\frac{r_{j}Y_{ij}}{r_{i}+r_{j}}+\frac{r_{i}\psi ( \mu _{i},\sigma _{i}^{2}) -r_{j}\psi ( \mu _{j},\sigma _{j}^{2}) -\log \lambda }{r_{i}+r_{j}}\text{.} \end{aligned}$$

Individual $i$’s certainty equivalent of consuming $\tilde{Y} _{ij}-c_{ij}-\psi ( \mu _{i},\sigma _{i}^{2})$ is (using the assumptions CARA + normal distribution):

$$\begin{aligned}&\frac{r_{j}}{r_{i}+r_{j}}(\mu _{i}+\mu _{j})-\frac{r_{j}}{2}\Big ( \frac{r_{j}}{r_{i}+r_{j}}\Big ) ^{2}(\sigma _{i}^{2}+\sigma _{j}^{2})-\psi ( \mu _{i},\sigma _{i}^{2}) \nonumber \\&\quad +\frac{r_{i}\psi ( \mu _{i},\sigma _{i}^{2}) -r_{j}\psi ( \mu _{j},\sigma _{j}^{2}) -\log \lambda }{r_{i}+r_{j}}, \end{aligned}$$

(12)

and individual $j$’s certainty equivalent of consuming $c_{ij}-\psi ( \mu _{j},\sigma _{j}^{2})$ is

$$\begin{aligned}&\frac{r_{i}}{r_{i}+r_{j}}(\mu _{i}+\mu _{j})-\frac{r_{i}}{2}\Big ( \frac{r_{i}}{r_{i}+r_{j}}\Big ) ^{2}(\sigma _{i}^{2}+\sigma _{j}^{2})-\psi ( \mu _{j},\sigma _{j}^{2}) \nonumber \\&\quad -\frac{r_{i}\psi ( \mu _{i},\sigma _{i}^{2}) -r_{j}\psi ( \mu _{j},\sigma _{j}^{2}) -\log \lambda }{r_{i}+r_{j}}\text{.} \end{aligned}$$

(13)

(12)+(13) gives

$$\begin{aligned} C_{ij}=(\mu _{i}+\mu _{j})-\frac{R(\sigma _{i}^{2}+\sigma _{j}^{2})}{2} -[\psi ( \mu _{i},\sigma _{i}^{2}) +\psi ( \mu _{j},\sigma _{j}^{2}) ]. \end{aligned}$$

One can also refer to Schulhofer-Wohl (2006), who provides the general results for ISHARA preference, with CARA as a special case.

Appendix B: Modeling endogenous risks as portfolio choice

In this section, we model the endogenous risks as a portfolio choice. The game is similar to that in Di Cagno et al. (2011). In particular, each agent is endowed with 50 cents. At period 0 each agent voluntarily matches with a mate from the opposite side. A couple hence has total wealth of one dollar. Each couple $(i,j)$ will commit to invest $1-\alpha \in [0,1]$ dollars in a risk free asset with gross return $\mu _{1}$ and $\alpha$ in a normally distributed risky asset with mean $\mu _{2}$ and variance $\sigma ^{2}$. Assume $\mu _{2}>\mu _{1}$. The agents will also commit to rules for sharing their income in each state of the world. At period 1, agents invest according to the committed actions. Then, the value of all shocks are realized and agents consume according to the prior sharing rules.

For a matched couple, they will choose $\alpha$ to maximize the household certainty equivalent:

$$\begin{aligned} \max _{\alpha }\left( 1-\alpha \right) \mu _{1}+\alpha \mu _{2}-\frac{R}{2} \alpha ^{2}\sigma ^{2}\text{.} \end{aligned}$$

Taking F.O.C. gives

$$\begin{aligned} \alpha =\frac{\mu _{2}-\mu _{1}}{R\sigma ^{2}}. \end{aligned}$$

(14)

Notice that $\alpha$ is increasing in the risk premium $\mu _{2}-\mu _{1}$, and decreasing in the riskiness of the risky asset and the representative agent’s degree of risk aversion. The household certainty equivalent $C$ is a function of the representative agent’s coefficient of risk aversion $R$. By Proposition 3, NAM arises if an only if

$$\begin{aligned} \frac{\partial ^{2}C_{ij}^{*}}{\partial r_{i}\partial r_{j}}=C_{ij}^{*\prime \prime }\left( R\right) \frac{R}{2}\frac{\partial ^{2}R}{\partial r_{i}\partial r_{j}}+C_{ij}^{*\prime }\left( R\right) \frac{\partial ^{2}R}{\partial r_{i}\partial r_{j}}\le 0. \end{aligned}$$

(15)

By the Envelope Theorem, we have $C^{\prime }( R) =-\frac{\alpha ^{2}\sigma ^{2}}{2}<0$, and $C^{\prime \prime }( R) =-\alpha \sigma ^{2}\frac{\partial \alpha }{\partial R}>0$: $C$ is decreasing and convex in $R$. Again, here we have two opposite effects. The second term $C_{ij}^{*\prime }( R) \frac{\partial ^{2}R}{\partial r_{i}\partial r_{j}}$ reflects the standard risk-sharing effect for exogenous $\alpha$. This term is negative and drives the matching pattern to be negatively assortative. The first term is due to the fact that agents can optimally choose $\alpha$ and hence it reflects the risk-management effect. $C$ is convex in $R$, so this term is positive. The risk-management effect drives the matching pattern to be positively assortative.

Substituting $C^{\prime }( R) =-\frac{\alpha ^{2}\sigma ^{2}}{2}$, $C^{\prime \prime }( R) =-\alpha \sigma ^{2}\frac{\partial \alpha }{\partial R}$ and (14) into (15), we get

$$\begin{aligned} \frac{\partial ^{2}C_{ij}^{*}}{\partial r_{i}\partial r_{j}}\equiv 0\text{.} \end{aligned}$$

A less risk-averse agent prefers a highly risk-averse agent who pays a high risk premium. But then the collective investment in the risky asset is much less than his preferred amount. The risk-sharing effect and the risk-management effect offset each other. Hence, the matching pattern is arbitrary. That the two effects offset each other depends crucially on our specific assumption of CARA preference and normal distribution. But again, we see that the equilibrium matching pattern balances the trade-off between risk sharing and risk managing. We leave it to future work to explore the case of general preferences and general distributions.