Assortative matching and risk sharing

Abstract

This paper explores the sorting patterns in a two-sided matching market where agents facing different risks match to share them. When preference belongs to the class of harmonic absolute risk aversion (HARA), the risk premium is perfectly transferable within each partnership; thus a stable match minimizes the social cost of risk. In the systematic risk model, where agents are ranked by their holdings of a common risky asset, the convexity of the joint risk premium in joint risk size leads to negative assortative matching (NAM). In the idiosyncratic risk model, where agents are ranked by their independent riskiness in the sense of second-order stochastic dominance (SSD), NAM arises when preference exhibits decreasing absolute risk aversion (DARA) in the sense of Ross and riskier background risk leads to more risk-averse behavior. However, NAM may fail to arise when riskier background risk leads to more risk-tolerant behavior.

Introduction

When insurance and financial markets are incomplete, individuals often form partnerships to diversify their risks. For instance, families – mainly in developing countries – often arrange for long-distance marriages for the purpose of sharing production shocks, manufacturing employers often cushion temporary shocks on profit by sharing with their workers, and different parties in related businesses sometimes develop joint ventures to share resources and revenues for mutual benefit (Rosenzweig and Stark, 1989, Townsend, 1994, Fafchamps and Lund, 2003, Bigsten et al., 2003). When risk sharing is a primary concern in forming partnerships, it is legitimate to ask how the agents should match to insure against risks. Do the evidences in the marriage market or the financial market reflect the mitigation of an incomplete insurance market, or are they boosted by other concerns at the cost of efficiency in risk sharing?

In this paper, we examine the sorting patterns in a two-sided matching market where agents facing different risks match to share them. It is known that when agents have different degrees of risk aversion, negative assortative matching (NAM) arises because risk bearings are generally substitutes: a very risk-averse female is a demanding buyer for insurance and a very risk-tolerant male is a ready seller for it (Chiappori and Reny, 2006, Schulhofer-Wohl, 2006, Legros and Newman, 2007). Rather than employing different degrees of risk aversion, our paper focuses on different risks that each agent faces. Since the Pareto frontier in a given match does not have constant slope, standard type-complementarity conditions (Becker, 1973) cannot be used in general. However, with respect to risk-sharing problems, it is known that when preference belongs to the class of harmonic absolute risk aversion (HARA), the Pareto frontier in the monetary-equivalent space is a straight line, or, in other words, the total surplus summarized by the certainty equivalent is independent of how risk sharing is performed. In this case, the matching game permits a transferable expected utility representation and the type-complementarity condition translates into minimizing social risk premium.

We then consider two applications: one where risks are perfectly correlated and one where risks are independent. In the systematic risk model, agents are ranked by their percentages of ownership of a common risky asset. Because joint risk premium is a convex function of the joint size of the common risk, it is extremely costly to pair two highly risky agents together. Hence, negative sorting is socially preferable and stable. One may wonder to what extent the result of negative sorting depends on the HARA assumption. As a robustness check, we show that, with general utility functions, NAM still arises if the supports of all risks are not too large compared with agents' risk-free incomes and/or if risk tolerance is sufficiently linear.

In the idiosyncratic risk model, agents are ranked by their independent riskiness in the sense of second-order stochastic dominance (SSD). NAM arises if the preference exhibits decreasing absolute risk aversion (DARA) and if riskier background risk leads to more risk-averse behavior, but may fail to arise when riskier background risk leads to more risk-tolerant behavior. There are four key points to note here. First, the conditions for NAM have clear economic implications and are supported by empirical evidence. Guiso et al. (1996) concluded from Italian survey data that a consumer's perception of a riskier distribution of uninsurable human-capital wealth is negatively related to the proportion of risky assets held in his/her investment portfolio. Second, the seemingly strong conditions for NAM to arise come from the fact that we are looking for the equilibrium sorting patterns for any SSD-ordered risks. For a special case of the SSD order where risks are ranked in the sense of SSD by taking the form of adding independent noise, we only need HARA and DARA to guarantee NAM. Third, when risks are large with respect to agents' risk-free incomes, an SSD deterioration in the background risk may lead to more risk-tolerant behavior, and thus, NAM may fail to arise in equilibrium. Fourth, the different results in the two applications suggest that one should investigate carefully whether agents are sharing highly correlated risks or independent risks.

The results of this paper may help us to understand the composition of risk-sharing groups in developing countries. Ghatak (1999) argued that positive assortative matching (PAM) should arise because similar people will find it easier to monitor and enforce informal contracts. Empirical evidence, however, is mixed: on one hand, Bacon et al. (2014) found evidence that individuals tend to form partnerships with others having a similar risk attitude; and Arcand and Fafchamps (2012) also found solid evidence of positive sorting for peers with respect to physical or ethnic proximity as well as wealth or household size. On the other hand, Dercon et al. (2006) found little evidence of positive sorting in group-based funeral insurance. Our results from the idiosyncratic model suggest that the risk-sharing effect might drive matching to be negative assortative and, therefore, offset the monitoring and enforcing effects; however, when risks are large compared with individuals' risk-free incomes, it is possible that the two effects might work in the same direction and drive matching to be positive assortative.

Our work contributes to the recent literature on risk-sharing matching games. Since the efficient risk sharing rule is typically nonlinear, the risk-sharing matching game permits non-transferable utilities, and thus, standard type-complementarity conditions cannot be used. Legros and Newman (2007) noticed that the risk-sharing matching game admits a transferable utility representation when agents have logarithmic or exponential utility functions. Schulhofer-Wohl (2006) generalized their findings, showing that the game admits a transferable utility representation when preferences are in the harmonic absolute risk aversion class with identical shape (ISHARA). Both Legros and Newman (2007) and Schulhofer-Wohl (2006) proved that the equilibrium sorting pattern is negative assortative on risk preferences. Chiappori and Reny (2006) further showed that negative sorting over risk preferences is robust under general utility functions. The key difference between our work and the existing literature is that our paper focuses on different risks that each agent faces rather than different degrees of risk aversion. Among the papers on risk-sharing matching games, ours is among the first to investigate sorting over agents' risk exposure.¹

There are two reasons we think examining riskiness is important. First, individual risk preferences have not proved to be stable across different stimulus domains and situations. For example, the predictive power of investors' risk taking heavily depends on whether their risk attitudes are elicited in an investment-related context (Slovic, 1964; MacCrimmon and Wehrung, 1986, MacCrimmon and Wehrung, 1990; Schoemaker, 1990, Schoemaker, 1993; Weber and Milliman, 1997). Second, because income riskiness is presumably easier to observe than attitudes toward risk, one might expect to drive testable predictions concerning the role of risk-sharing in the formation of partnerships more easily if agents are ranked on the basis of riskiness.

Moreover, the results of our paper differ from those in the literature. Legros and Newman (2007), Schulhofer-Wohl (2006) and Chiappori and Reny (2006) rigorously proved that NAM arises if agents hold the same exogenous risky assets but differ in their risk attitude. Following their results, Li et al. (2013) showed that PAM may arise if agents' incomes are endogenous (also see Wang, 2013a). Wang (2013b) showed that the presence of moral hazard may also lead to PAM. Our results show that without any other confounding factors such as endogenous income or moral hazard, the counter-intuitive PAM may arise if agents differ in their idiosyncratic risks instead of their risk preferences: while agents with highly risky assets always try to avoid matching with other large, perfectly-correlated risks, they might prefer to match with other large, independent risks.

The rest of this paper is organized as follows. Section 2 presents the risk-sharing matching game. Section 3 applies a monotonic transformation to this game and characterizes the stable match. Sections 4 and 5 consider two applications, one where risks are perfectly correlated and the other where risks are independent. Section 6 extends the model to allow individuals to have different incomes and face different risks. Section 7 concludes the paper.