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  • Ambiguity and Asymmetric Information

    Rachel J. Huang Graduate Institute of Finance

    National Taiwan University of Science and Technology

    Arthur Snow Department of Economics

    University of Georgia

    Larry Y. Tzeng Department of Finance

    National Taiwan University

    Preliminary version

    Abstract The theoretical literature has found that advantageous selection could be observed in equilibrium when there is multi-dimensional heterogeneity of customers or both hidden action and hidden information are employed. This paper proposes a third approach for advantageous selection. Based on the classic Rothschild and Stiglitz (1976) model, we consider ambiguity-averse individuals who face a general risk and a specific risk in a perfectly competitive insurance market. The individuals have private information regarding the specific risk but have unbiased ambiguous beliefs regarding the general risk. Individuals' preferences are characterized by Klibanoff, Marinacci and Mukerji's (2005) smooth model, and they are homogeneous except for their risk type. We find that both pooling and separating equilibria can exist. Furthermore, we find that when the single crossing property does not hold, equilibrium could be determined based on adverse selection or advantageous selection. Keywords: advantageous selection, adverse selection, ambiguity. JEL classification: D80, G22, C30

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    Ambiguity and Asymmetric Information

    1. Introduction

    In Rothschild and Stiglitz's (1976) classic adverse selection model, high risk

    types of individuals self-select a contract with higher coverage, whereas the low risk

    types choose a contract with lower coverage in equilibrium. Their model thus predicts

    a positive correlation between insurance coverage and accident occurrence. The

    empirical evidence regarding the positive relationship is, however, mixed. In acute

    health insurance and annuity markets, researchers have found empirical evidence that

    is consistent with the prediction of adverse selection.1 On the contrary, a significant

    negative correlation is supported in the markets for life insurance (Cawley and

    Philipson 1999, McCarthy and Mitchell 2003), long-term care (Finkelstein and

    McGarry 2006), medigap insurance (Hurd and McGarry 1997, Fang et al. 2006),

    reverse mortgages (Davidoff and Welke 2007) and commercial fire insurance (Wang,

    Huang and Tzeng 2009).

    This phenomenon of a negative correlation between insurance and claim

    frequency is documented as "advantageous selection". To provide theoretical support

    for advantageous selection, the researchers have adopted the following two different

    approaches. The first one is to consider the multi-dimensional heterogeneity of

    customers. Liu and Browne (2007) exogenously assume that individuals are

    heterogeneous with respect to risk type and risk aversion and find advantageous

    selection in equilibrium when the insurance is not fair.2 Netzer and Scheuer (2010)

    endogenize heterogeneity in wealth levels and assume that both risk type and patience

    are private information. They show that a negative correlation between insurance

    1 For health insurance, see Cutler and Zeckhauser (2000). Mitchell et al. (1999), Finkelstein and Poterba (2004) and McCarthy and Mitchell (2003) respectively examine the annuity market in the U.S., the U.K. and Japan. 2 Smart (2000), Wambach (2000), Villeneuve (2003) also assume that risk type and risk aversion are private information. Those models predict a positive correlation as in Rothschild and Stiglitz (1976).

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    coverage and risk type can be obtained.

    The second approach is to integrate both hidden action and hidden information.

    Individuals are homogeneous except for their degree of risk aversion (de Meza and

    Webb 2001), patience (Sonnenholzner and Wambach 2009), overconfidence (Huang,

    Liu and Tzeng 2010), or regret (Huang, Muermann and Tzeng 2011). This

    one-dimensional heterogeneity of customers will affect the optimal choice of the

    hidden action regarding self-protection and further cause heterogeneity in the risk

    type. They respectively show that advantageous selection can appear in equilibrium

    since risk-neutral, impatient, overconfident and regretful customers might spend less

    on insurance and prevention, thereby becoming higher risk types.

    In this paper, we propose a third approach by adopting ambiguity. Ambiguity

    aversion describes the situation in which individuals prefer a lottery with a certain as

    opposed to an uncertain probability even though the mean of the uncertain probability

    is the same as that of the certain probability. To employee ambiguity, we first assume

    that the risk of an individual can be decomposed into two parts: a general risk and a

    specific risk. For example, the car accident probability of an individual depends on

    the individual's driving skills (specific risk) and the traffic condition (general risk).3

    The mortality rate is jointed determined by the health status (specific risk) and the

    lethality of diseases (general risk).

    We further assume that the specific risk is private information to the individuals

    but the general risk is ambiguous to all participants in a perfectly competitive

    insurance market. All participants in the market share the same ambiguous beliefs

    regarding the general risk.4 In addition, we assume that individuals are

    3 Huang, Tzeng and Wang (2012) show that a driver's car accident rate is positively significantly affected by both the driver's kilometer driven and the average kilometer driven of the society. The former could be viewed as a specific risk, whereas the later could be viewed as a general risk. 4 Our setting is close to but different from that of Seog (2009). Seog (2009) assumes that the general risk is privately known by the insurers and the individuals have precise beliefs regarding the general

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    ambiguity-averse and are characterized by the same ambiguity preferences. In other

    words, individuals are heterogeneous only in the specific risk. Note that we do not

    assume that individuals can make any effort to change the loss probability. Thus, our

    model is neither one that follows a multi-dimensional approach nor a model with both

    hidden action and hidden information.

    To model ambiguity aversion, we adopt Klibanoff, Marinacci and Mukerji's

    (2005) smooth model.5 In their model, the ambiguity function captures the utility of

    an ambiguity-averse individual and is set as a concave transfer of the individual's

    traditional expected utility. Thus, under ambiguous beliefs, utility could be presented

    as the expected ambiguity function over the ambiguous beliefs. This two-stage

    decomposition of the decision process helps us to apply the well-developed

    techniques under the expected utility framework to the analysis of problems involving

    ambiguity aversion.

    We find that a pooling contract might be optimal although the equilibrium occurs

    only for very special parameter values. Furthermore, as in Rothschild and Stiglitz

    (1976), we find that the market might settle on separating equilibrium. However, the

    equilibria no longer exhibit a monotone relationship between insurance coverage and

    risk in our model. The equilibrium might thus settle on adverse selection or

    advantageous selection. It depends on insurance loading and how the degree of

    ambiguity aversion varies when the expected utility increases.

    Our paper contributes to the literature in the following ways. First, we propose a

    different approach to advantageous selection. Second, to the best of our knowledge,

    our paper is the first one to examine the effect of ambiguity aversion on competitive

    risk. In other words, ambiguity is absent in his model. 5 Since Ellsberg (1961), researchers have proposed different settings to characterize ambiguity aversion preferences. For example, see Gilboa and Schmeidler (1989), Schmeidler (1989), Ghirardato, Maccheroni, and Marinacci (2004), and Klibanoff, Marinacci and Mukerji (2005).

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    insurance markets where the individuals have private information. There are several

    papers in the literature that incorporate ambiguity in adverse selection problems. For

    example, see Jeleva and Villeneuve (2004), Chassagnon and Villeneuve (2005) and

    Vergote (2010). However, they all focus on a monopolistic rather than a competitive

    insurance market. Third, our paper is the first one to adopt Klibanoff, Marinacci and

    Mukerji's (2005) smooth model and to apply it to insurance markets under

    asymmetric information.

    The remainder of this paper is organized as follows. Section 2 introduces the

    demand for insurance in the presence of ambiguity under full information. Section 3

    employees asymmetric information and examines the equilibrium. Section 4

    concludes the paper.

    2. Full information: Demand for insurance in the presence of ambiguity

    To prepare the background knowledge, we first examine the demand for

    insurance in the presence of ambiguity under full information. In a perfectly

    competitive insurance market, individuals face a binomial property risk with either a