Loss Aversion Leading to Advantageous Selection

Transcription

1 Loss Aversion Leading to Advantageous Selection Christina Aperjis and Filippo Balestrieri HP Labs [This version April 211. Work in progress. Please do not circulate.] Abstract Even though classic economic theory predicts that ex-post risk and coverage are positively correlated in an insurance market i.e., people that bought more insurance are more likely to have a bigger damage), the opposite has been observed in a variety of settings. From the prospective of the insurer, this puzzle is associated with advantageous selection, because the agents who buy insurance are also the cheapest to insure in the market. Previous literature has attempted to explain advantageous selection in settings with moral hazard. In this paper, we consider a model with loss-averse agents without moral hazard) and we offer plausible conditions under which advantageous selection occurs. These are based on the relation between the distributions of losses and reference points. We show that the willingness to pay for an insurance product increases with the value of the reference point. Indeed, in a loss-aversion setting, an insurance product is valuable to an agent if it provides a hedge against the risk of suffering losses below the reference point. Under certain conditions, when agents with higher expected losses have lower reference points, advantageous selection arises. 1 Introduction Empirical works on insurance markets brought to light the so called insurance puzzle. The puzzle is based on the observation that, in several markets, there is a negative correlation between insurance coverage and ex-post risk. In other words, people who buy insurance experience less damages than people who do not buy insurance. We thank Bengt Holmstrom, Sergei Izmalkov, Florian Schuett, Julie Ward, Richard Zeckhauser, and seminar participants at IIOC 211 for their comments. 1

2 Even though not ubiquitous, evidence of the insurance puzzle has been found in several insurance markets. Examples include long term care insurance [Finkelstein and McGarry, 26], car insurance [Chiappori et al., 26], and credit card insurance [de Meza and Webb, 21]. From the prospective of the insurer, instead of the adverse selection predicted by economic models, the insurance puzzle is associated with advantageous or favorable selection, according to which the agents who buy insurance are also the cheapest to insure in the market. Advantageous selection is inconsistent with standard economic theory, according to which buyers who are more likely to incur losses should have higher willingness to pay for an insurance product [e.g., Rothschild and Stiglitz, 1976]. In this work, we show that a behavioral model of loss-aversion provides a natural explanation for the insurance puzzle. In a loss-aversion framework, buyers with higher expected losses tend to have lower reference points. We show that the willingness to pay for an insurance product increases with the value of the reference point used by a loss averse buyer. Then, advantageous selection arises under certain conditions). We consider a setting with a monopolist insurer and agents with private information about their probability distributions over losses. For the sake of simplicity, we consider a stylized model with two types of agents: a bad type, with a high expected loss; and a good type, with a low expected loss. Each agent decides whether to fully hedge against the risk of incurring losses by buying an insurance product. We study this decision problem in the context of two frameworks, risk-aversion and loss-aversion. We identify the conditions under which the insurance puzzle arises and compare the two frameworks through these conditions. First, we consider the risk aversion framework. We examine different settings in terms of the distribution of losses. We show that, in many settings, the insurance puzzle may be explained only assuming a specific correlation between the degree of risk aversion and the types of the agents: the agent who is more risk averse needs to be also the one with the lowest expected loss. We then examine the loss-aversion framework. We model each agent s utility as a summation of a consumption term and a gain-loss term. The gain-loss term increases or belittles the utility depending on how the consumption level compares to a reference point. If the actual consumption is more than the reference, the utility increases; if it is less, the utility decreases. Loss aversion reflects the fact that, as verified in several experimental studies, individuals show greater aversion for losses than appreciation for gains [e.g., Kahneman and Tversky, 1979]. In other words, people are more motivated by avoiding a loss than acquiring a similar gain. To define the reference point, we follow Kőszegi and Rabin [26] and assume that a different reference point is associated with every action. The advantage of this approach is that the reference points are endogenized. The solution concept we use to study the decision problem of each agent is the personal equilibrium; informally, an action is a personal equilibrium if it maximizes the agent s utility when he expects to play this action, that is, when the reference point is the one 2

3 associated with this action. When both buying and not buying insurance are personal equilibria for an agent, we assume that the agent will select the action that is associated with the highest ex ante expected utility; this is the concept of a preferred personal equilibrium [Kőszegi and Rabin, 26]. A reference point depends on the consumption level induced by the action it corresponds to. If an action brings a deterministic level of consumption, then the corresponding reference point is equal to that consumption level. On the other hand, if an action is associated with a random outcome, a variety of potential reference points arises; the reference point could be the expected consumption level, some other statistical function of the agent s random consumption level or the random consumption level itself [Kőszegi and Rabin, 26, Schmidt et al., 28]. In an insurance market, the action of not buying insurance is associated with a random outcome, since the agent remains exposed to the risk of incurring losses. In order to study the role of reference points, we focus on a general deterministic reference point throughout the paper and consider the case of a stochastic reference point in the appendix. We expect agents with larger loss distributions to associate lower reference points with the action of not buying insurance. 1 We show that when a lower reference point is associated with not buying insurance, the agent is willing to pay less for insurance. Thus, the distribution of losses may have an indirect effect on the agent s willingness to pay through the reference point, according to which agents with higher losses are willing to pay less for insurance. In particular, an insurance product that provides a hedge against losses becomes less appealing to agents with higher losses, because the cases in which the insurance allows them not to fall below their reference point and suffer the extra disutility connected with the loss feeling) are more limited. More generally, our model allows us to consider the role of reference points as making insurance more or less attractive to the agent. In the loss aversion framework, the distribution of losses has two effects on the agent s willingness to pay for insurance. On one hand, there is the indirect effect through the reference point, according to which higher losses decrease the agent s willingness to pay for insurance. On the other hand, there is also the direct effect of standard adverse selection, according to which higher losses increase the agent s willingness to pay for insurance. When the indirect effect through the reference point dominates, the good type is willing to pay more than the bad type for the insurance product. Advantageous selection arises when the willingness to pay of the good type is sufficiently larger than the willingness to pay of the bad type so that the seller obtains a higher profit from selling only to the good type than selling to both types. We show that the insurance puzzle can be represented in a wider range of settings with the loss aversion framework than with the risk aversion framework. This is possible without imposing any condition on the heterogeneity in the degrees of loss aversion across agents. We consider specific cases for which 1 This arises with the stochastic reference points suggested by Kőszegi and Rabin [26] and Schmidt et al. [28]. 3

4 advantageous selection is not possible with risk aversion, but is possible with loss aversion. 2 Previous literature has tackled the insurance puzzle by providing theoretical explanations in contexts of moral hazard and heterogeneity in the degree of risk aversion [de Meza and Webb, 21, Jullien et al., 27]. In these papers, ex-ante all consumers are assumed to be exposed to the same probability distribution over losses, but in equilibrium, consumers who are more risk averse ergo, more willing to buy an insurance product) are also the ones who spend more effort to minimize the probability of a damage. On the other hand, Netzer and Scheuer [21] consider a setting with heterogeneity in risk and patience with dynamic accumulation of wealth. In this paper, we address two issues that the previous literature leaves open. First, we determine conditions under which the insurance puzzle arises in the absence of moral hazard. This shows that the insurance puzzle can arise in markets where customers cannot or are not willing to increase the probability of incurring a loss with their behaviors. 3 Second, in contrast to previous literature, we show that advantageous selection can appear in a market with homogeneous agents in terms of the degree of loss aversion. The use of behavioral models to explain puzzling consumers choices when facing uncertainty is a promising field for investigation. Here we make an attempt with respect to loss-aversion and the insurance puzzle. Loss aversion seems to provide a natural explanation based on the relation between reference points and expected losses. The rest of the paper is organized as follows. The model is set up in Section 2. Section 3 analyzes advantageous selection in a setting with risk aversion. In Section 4, we consider loss aversion: we present different reference point specifications and derive conditions for advantageous selection. Section 5 concludes. In the Appendix we consider a setting with loss aversion and stochastic reference points a la Kőszegi and Rabin [26]). 2 Model We consider an environment with two agents that are exposed to the risk of incurring losses, and a seller that is offering them an insurance to hedge against that risk. Each agent i {1, 2} is described by a random variable X i. Each realization of X i, X i = x, represents a monetary loss for agent i. We denote the set of all the possible losses that each agent i may suffer by Ξ. Let F i x) be the cumulative distribution function associated with the random variable X i. We 2 We consider the same setting as Rothschild and Stiglitz [1976]. 3 The latter is particularly relevant for certain insurance markets, such as that of life insurance, in which the insurance puzzle has been observed. In particular, even though an agent s behavior does affect the probability of dying, we would not expect an agent to start smoking just because he has life insurance; Cutler and Zeckhauser [2] discuss this issue for health insurance. 4

5 do not make any restrictive assumptions on X i ; our results hold for both discrete and continuous X i. In the absence of insurance, when agent i suffers a loss x, then his consumption is c = w x, where w denotes the level of initial wealth, and his utility is u i c) = u i w x). We assume that the initial wealth w is the same for all agents and that the utility function of each agent i is increasing in the level of consumption. We define t i = {X i } as the type of agent i and we assume that each agent is privately informed about his type. In other words, agents are privately informed about their loss distributions. We call good bad) the type with the lowest highest) expected loss E X i ). Throughout the paper, we consider only cases in which agents have different types. In that sense, the words agent and type become interchangeable. We designate agent i = 1 to be the good type, thus E X 1 ) < E X 2 ). A seller S offers insurance for a price p. The insurance is a contract that obligates the seller the insurer) to fully compensate any buying agent the insuree) for any loss suffered. We assume that the seller is risk neutral. This entails that, when an insuree incurs a loss x, the seller s utility and consumption is u S c) = c = p x. When agent i does not buy the insurance product, the seller s utility is zero i.e., u S = ), and the agent s expected utility is E u i w X i )). When agent i buys the insurance product, the seller s expected utility is E [ u S p, X i ) ] = p E X i ), agent i s consumption is c = w p, and his utility is u i w p). In this model the timing of the interaction between seller and agents is the following. First, the seller sets a price for the insurance. Then, the agents simultaneously learn the price and each agent decides whether to buy insurance. The equilibrium of the game is Bayes-Nash and is a set of price and actions. The price is the price p set by the seller, the actions are the actions of buying insurance or not buying insurance chosen by agent 1 and agent 2. We assume that each agent is rational: he decides to buy the insurance whenever the utility from buying is higher than the expected utility of not buying. We assume that the seller is rational: she sets a price p such that her expected utility is maximized. We define equilibrium with advantageous selection any equilibrium in which the good type buys insurance, the bad type does not, and the seller maximizes her profit π. An equilibrium with advantageous selection corresponds to an example of the insurance puzzle. Notice that the same setting that we have introduced here to describe the insurance purchasing behavior of different agents each indexed by i) in one environment, can also be used to represent the behavior of one agent in different environments each indexed by i). In light of this second interpretation, an equilibrium with advantageous selection would correspond to an inconsistency in the choices of a given customer offered to buy insurance in different situations and ending up buying coverage when his expected loss is lower. In the next sections we consider different specifications for the utility functions u i and different type spaces. For each scenario, we identify the conditions 5

6 under which advantageous selection may arise. 3 Risk Aversion As benchmark, we first consider a scenario in which the agents are risk averse. This entails that u i c) is a concave function for each i. Observe that agent i obtains utility u i w p) from buying the insurance and E[u i w X i )] from not buying. Previous literature has identified conditions for advantageous selection in settings where agents are risk averse. In these models the good type the type with the lowest expected loss) is the one with the highest degree of risk aversion. Such negative correlation between expected loss and degree of risk aversion is endogenous. Agents control their probability of incurring losses exercising an effort. In equilibrium, highly risk averse agents exercise more effort. The negative correlation between expected loss and degree of risk aversion may be a natural phenomenon in models with moral hazard. However, this is not necessarily the case in settings of pure information asymmetry, where any combination of risk aversion degree and loss distribution is plausible. Is negative correlation between expected loss and degree of risk aversion a necessary condition for equilibria with advantageous selection to arise? Can we still use a risk aversion model and get advantageous selection even with no negative correlation between expected loss and degree of risk aversion? 3.1 Advantageous Selection We consider here agents with the same degree of risk aversion the same utility function) and we derive the conditions that support equilibria with advantageous selection. Notice that the assumption of different types having the same degree of risk aversion also provides a setting to analyze the behavior of one agent in different environments. Suppose that both agents have the same utility function u 1 = u 2 = u. Then, agent i is willing to buy insurance if and only if E[uw X i )] uw p). Thus, agent i s willingness to pay 4 for insurance is p i w u 1 Euw X i ))). If the bad type is willing to pay at least as much as the good type for insurance, then advantageous selection cannot occur. Suppose that p 2 < p 1, that is, the good type is willing to pay more than the bad type for insurance. Then, the seller can either set the insurance price equal to p 1 and as result only sell to the good type, or set the insurance price at p 2 and sell the insurance to both 4 Throughout the paper, the term willingness to pay refers to the maximum amount that the agent is willing to pay for insurance. 6

7 agents. The expected profit from selling only to agent 1 is p 1 EX 1, whereas the expected profit from selling to both agent is 2p 2 EX 1 EX 2. We conclude that advantageous selection will occur in equilibrium if and only if p 1 + EX 2 2p 2. Moreover, if p 1 + EX 2 > 2p 2, this is the unique equilibrium, that is, there does not exist an equilibrium at which the seller sells insurance to both agents. In what follows, we verify whether advantageous selection arises given different type spaces. The type of an agent has been defined as the random variable X i. When we consider different utility functions u i, agent i s type is redefined as the tuple {X i, u i }. The set of possible realization x of X i=1,2 is Ξ. 3.2 Examples We first assume that Ξ is a set of two elements: Ξ = {, d}, where d >. In other words, there are only two possible outcomes: loss or no loss, and the size of the loss is the same for all agents. Agents differ only in terms of their probability of incurring the loss. This is the same type space considered by Rothschild and Stiglitz [1976]. Let q i denote the probability agent i incurs a loss d. Given that agent 1 is the good type, then, by definition of good type, q 1 < q 2. Proposition 1. Suppose that u 1 u 2 u and Ξ = {, d}. Then, there do not exist equilibria with advantageous selection. Proof. Assuming that q 1 < q 2, a necessary condition for advantageous selection is that p 1 > p 2, which can only hold if E[uw X 1 ))] < E[uw X 2 ))]. For Ξ = {, d}, this condition can be rewritten as 1 q 1 ) u w) + q 1 u w d) < 1 q 2 ) u w) + q 2 u w d). However, this cannot hold since q 1 < q 2 and u is increasing. We conclude that advantageous selection cannot occur. This implies that in order to have an equilibrium with advantageous selection we need the agents to have different utilities. Indeed, there exist functions u 1 and u 2 for which advantageous selection occurs with Ξ = {, d}. The following discussion shows that the good type has to be more risk averse in this case. In particular, consider a setting with Ξ = {, d} and utilities u 1 and u 2. The following conditions are necessary for advantageous selection: 1 q 1 ) u 1 w) + q 1 u 1 w d) u 1 w p) ; u 2 w p) 1 q 2 ) u 2 w) + q 2 u 2 w d). 7

8 The first condition implies that agent 1 buys insurance at price p; the second that agent 2 does not buy insurance at price p.) Since q 1 < q 2 and u 2 is increasing, the latter condition implies that u 2 w p) < 1 q 1 ) u 2 w) + q 1 u 2 w d). Thus, when deciding between a deterministic consumption level w p and a lottery which gives w d with probability q 1 and w with probability 1 q 1, agent 1 prefers the safe option and agent 2 prefers the lottery. Thus, agent 1 is more risk averse than agent 2 when choosing between w p and w X 1. Proposition 1 shows some limitations of risk aversion models in representing the insurance puzzle. For advantageous selection to arise, the agents need to show different degrees of risk aversion in a very specific manner, or, alternatively, the type space needs to be sufficiently complex. The negative correlation between expected loss and degree of risk aversion shown by previous works on risk aversion and moral hazard may just reflect the limitations of a risk aversion setting, especially in simple loss-no loss settings. 4 Loss Aversion In this section, we consider a model in which agents are loss averse. We study how a loss aversion framework accommodates advantageous selection and compare to the risk aversion framework of the previous section). Our goal is to identify conditions under which advantageous selection occurs. 4.1 Reference-Dependent Utility In a setting with loss aversion, the utility of a given consumption level is affected by how it compares to a reference point, with agents exhibiting greater aversion to losses than appreciation of gains. Following previous literature [Kőszegi and Rabin, 26, 27], we model the agents loss aversion through a utility function u i = u i c, r), that depends on two consumption levels: c, the actual consumption level, and r, the reference point-consumption level. We assume that each agent i s utility is a weighted sum of two terms: i) a consumption term, and ii) a gain-loss term, which depends on the difference between the actual consumption and the reference point. To isolate the consequences of loss aversion, we assume that i) the consumption utility is the identity function and ii) the gain-loss utility is piecewise linear. 5 More specifically, given a consumption level c and a reference r, agent i s utility is u i c r) = c + η µc r). The parameter η shows how much the agent weights the gain-loss term compared to the consumption term. 5 This is a standard approach, also taken by Kőszegi and Rabin [26, 27] 8

9 The gain-loss utility µ is given by, { x if x µx) = λx if x < where λ > 1. The parameter λ is the loss aversion coefficient and is assumed to be strictly greater than 1, so that µ is steeper for losses than for gains. A larger λ implies that the agent is more loss averse. Similarly to the case of risk aversion, different agents may show different degrees of loss aversion; such heterogeneity can be modeled assuming different values of η and/or λ. However, in this paper, we are going to assume homogeneity in loss aversion across agents. When agent i s consumption level is given by a random variable Y i and the reference point is r i, agent i s expected utility is 4.2 Equilibrium Concepts E[u i Y i r i )] = EY i + η E[µY i r i )]. In our model, an equilibrium consists of an insurance price set by the seller and actions i.e., buying or not buying insurance) chosen by the agents such that the seller maximizes her profit and the agents maximize their utilities. Notice that no strategic interaction between the agents is assumed: the utility of an agent is not affected by any other agent s action. Moreover, given that the insurance price is set in advance, we only need to consider agent i s decision problem in order to determine which action he selects in equilibrium. As mentioned, agent i selects the action that maximizes his utility. We define such an action as a personal equilibrium for agent i. In a loss aversion setting, an agent s utility is a function of both the consumption level and the reference point. Freedom in selecting the reference points may imply indeterminacy in identifying which action is selected: different reference points support different selections. Following Kőszegi and Rabin [26], we resolve this issue by endogenizing the reference points each agent s reference point becomes part of his personal equilibrium). This is done by modeling reference points as functions of the actions that the agent expects to take. For example, if agent i expects to take action a, then agent i s reference point becomes ri a. In equilibrium, each agent expects to take exactly the action he actually takes. In that way, in equilibrium, for each agent, the reference point and the selected action are consistent. Here we define the solution equilibrium) concepts of the agents decision problems that we will use in subsequent sections to study agents choices decisions) in an insurance market. 6 Let A i be a set of feasible actions of agent i, and a an element of such set. Let Yi a be the random) consumption associated with action a. That is, the 6 We note that the definitions given here can also be applied for more general utility functions than the ones used in this paper. 9

10 agent knows that if he selects action a, then his future consumption level will be given by Yi a and the realization of this random variable will occur at some time in the future. The reference point associated with action a and agent i is denoted by ri a, for each i. We assume that ra i depends on Yi a; for instance, ra i can be equal to the expected value of Yi a. We now give the definition of a personal equilibrium. Definition 1. An action a A i is a personal equilibrium PE) for agent i if, for every action a A i, E[u i Y a i r a i )] E[u i Y a i r a i )]. In words, an action a is agent i s PE if the agent s expected utility is maximized at a when he expects to select a, i.e., when his reference point is r a i. There may be multiple PE. Which one will the agent play? Since the agent has a single utility function, he can rank the PE outcomes in terms of ex ante expected utility. Then, following Kőszegi and Rabin [26], we assume that the agent will select the one that is associated with the highest ex ante expected utility. This is the concept of preferred personal equilibrium. Definition 2. An action a A i is a preferred personal equilibrium PPE) for agent i if it is a PE and for every action a A i that is a PE. E[u i Y a i r a i )] E[u i Y a i r a i )] In words, an action a is a PPE for agent i if it is a PE and the associated ex-ante utility is the highest in the set of all PE. A reference point depends on the consumption level induced by the action it corresponds to. If an action brings a deterministic level of consumption, then the corresponding reference point is equal to that consumption level. On the other hand, if an action is associated with a random outcome, a variety of potential reference points arises; the reference point could be the expected consumption level, a convex combination of the values in the support of the probability distribution over consumption levels or a random variable that represents the consumption level [Kőszegi and Rabin, 26, Schmidt et al., 28]. In this paper, we focus on general deterministic reference points. This allows us to study the role of reference points on agents decisions. In particular, we consider how agents decisions change as reference points change. Moreover, we analyze situations where the reference point is given by a specific statistic of the consumption level, such as the expected value or the median. On the other hand, the stochastic reference points that have been suggested in the literature do not allow such a comparison across different reference points. Nevertheless, we consider the case of stochastic reference points in the Appendix. In the specific context of the insurance market we introduced in Section 2, the following table summarizes the consumption levels and the reference points 1

11 that are associated with each action for each agent i. The parameter k i is assumed to be in the convex hull of the support of X i. a action) Yi a consumption level) ri a reference point) buy insurance w p w p not buy insurance w X i w k i Notice that different reference points are associated with different actions. Each reference point depends on the consumption level of the action it is associated with. The reference point associated with buying insurance is equal to w p, reflecting the expense p that an agent has to pay in order to buy insurance. Thus, the reference point associated with buying insurance is the same for all agents. The reference point associated with not buying insurance is w k i, where k i is a function of the random variable X i and may differ across different agents. We refer to k i as the reference loss level for agent i. We first consider the case in which the function k i = K i X i ) differs across agents and we assume that it is part of each agent i s private information. Then we analyze settings in which the function K is the same for all the agents and is common knowledge. 7 For instance, k i could be equal to EX i or to the mode of X i. In these cases, the only source of heterogeneity across the agents is their random consumption X i. Notice that, even when the function K is common knowledge, the fact that X i is private information of each agent i prevents the seller from knowing the reference point k i of each agent i. This implies that the seller cannot use this information to price discriminate across the agents. The value of k i can be interpreted as a threshold expressed in terms of consumption levels) that defines which consumption outcomes are perceived by agent i as losses and which ones as gains. Several factors e.g., psychological elements, past experiences, degree of care for what is at stake...) determine which outcomes bring a feeling of loss to the agent. Applying Definition 1, we find that it is a PE for agent i to buy if It is a PE to not buy if p EX i η E[µp X i )] 1) p EX η µk i p) η E[µk i X i )] 2) Now suppose that both buying and not buying are PE, that is, both 1) and 2) hold. Applying Definition 2, we find that it is a PPE to buy if whereas is a PPE to not buy if p EX i η E[µk i X i )], 3) p EX i η E[µk i X i )]. 4) 7 If k i = KX i ), then effectively the reference point is w KX i ). Alternatively, we could define the reference point to be Kw X i ). If K is linear as is the case when KX) = EX), the two approaches are equivalent. 11

12 In the following sections we will use 1)-4) to characterize equilibria and derive conditions under which advantageous selection occurs. We observe that each of these inequalities depends on either E[µp X i )] or E[µk i X i )]. Both of these quantities are the expected value of the gain-loss utility as a function of the difference between some constant which is either p or k i ) and the random variable X i. The following lemma shows that such an expectation depends on the area below the distribution function of the random variable X; this result will be useful for characterizing advantageous selection in the following sections. Lemma 1. Let X be a non-negative random variable and let F be its cdf. Then for any c >, E[µc X)] = λ 1) c F x)dx λc EX). Proof. First suppose that X is a continuous distribution with density fx). We first observe that E[µc X)] = On the other hand, c = c c x)fx)dx + λ c x)fx)dx + λ 1) = c EX λ 1) x c)fx)dx = = EX c + = EX c + c c x c)fx)dx c c c x)fx)dx c x c)fx)dx c c x)f x)dx F x)dx c x)fx)dx x c)fx)dx The last equality follows by integration by parts. We substitute the second equality in the first to conclude the proof. The proof for a discrete distribution is very similar. The first step is to show that E[µc X)] = c EX λ 1) x c)p[x = x i ]. Then, we observe that x i>c x i>cx c)p[x = x i ] = x i x c)p[x = x i ] x i<c = EX c + x i<cc x)p[x = x i ]. x c)p[x = x i ] 12

13 Finally, c x)p[x = x i ] = x i<c c because this is the area under the cdf F from to c. F x)dx, We now use Lemma 1 to rewrite the conditions for PE. Note that y + maxy, ) is the positive part of y. Lemma 2. i) Buying is a PE for agent i if and only if p EX i η λ 1 p F i x)dx. ii) Not buying is a PE for agent i if and only if ) η λ 1 ki F i x)dx k i p) + p EX i. Proof. i) follows from 1) and Lemma 1. We now show ii). By substituting 2) in Lemma 1, we see that not buying is a PE if and only if ki p EX i + ηλk i EX i ) η µk i p) ηλ 1) F i x)dx. 5) We now consider 2 cases. First, if k i p, then µk i p) = λp k i ) and the left hand side of 5) is equal to 1 + ηλ)p EX i ), so 5) can be rewritten as p EX i η λ 1 ki F i x)dx. 6) The second case is k i p. Then µk i p) = k i p and the left hand side of 5) is equal to p EX i + ηλk i ηλex i ηk i + ηp = 1 + ηλ)p EX i ) + ηλ 1)k i p), so 5) can be rewritten as p EX i η λ 1 We get the result by combining 6) and 7). ) ki F i x)dx k i p). 7) 13

14 4.3 Properties of Personal Equilibria This section gives some general properties of PE and PPE in an insurance market. Given that we focus on a single agent, we drop the index i. Our first result shows that a PE always exists, that is, for any instance either buying is a PE or not buying is a PE or both buying and not buying are PE. Proposition 2. A PE always exists. Proof. By Lemma 2, neither buying nor not buying is a PE if and only if η λ 1 p ) F x)dx < p EX < η λ 1 k F x)dx k p) +. However this cannot hold because k F x)dx k p) + p F x)dx. A consequence of Proposition 2 is that a PPE always exist. The following lemma shows a connection between PE and PPE depending on whether the price p is greater or small than the reference loss level k. These properties are used in the following section. Lemma 3. i) If p k and buying is a PE, then buying is a PPE. ii) If p k and not buying is a PE, then not buying is a PPE. Proof. i) If p k, then E[µp X)] E[µk X)]. Thus, 1) implies 3). That is, if buying is a PE, then buying is a PPE. ii) If p k, then µk p). Thus, 2) implies 4). That is, if not buying is a PE, then not buying is a PPE. The utility specification of the agents is such that they are at most risk neutral. Given that, whenever the price of the insurance is lower than the expected damage, buying is a PPE and in some cases the unique PE. This is context of the following lemma. Lemma 4. i) If p EX then buying insurance is a PE and a PPE. ii) If p < EX and p < k, then buying insurance is the unique PE. Proof. We first show that if p EX, then buying insurance is a PE. By Lemma 2 i), if p EX η λ 1 p F x)dx, 14

15 then buying is a PE. The right hand side is always positive. If p EX, then the left hand side is negative, so buying insurance is a PE. This shows the first part of i). We now show that if p < EX and p < k then not buying is not a PE, i.e., that 2) does not hold. Since p < k, µk p) = k p. On the other hand, E[µk X)] E[k X] = k EX. Thus, µk p) E[µk X)] k p) k EX) = EX p) >, which implies that 2) cannot hold when p < EX and p < k. This together with the fact that buying insurance is a PE when p EX implies ii). To conclude the proof for i), suppose that p EX and consider the following cases: If p < k, then buying is the unique PE, so it is a PPE. If p k, the fact the buying is a PE implies that it is a PPE by Lemma 3). We conclude that buying is a PPE for any k. 4.4 Willingness to Pay and Reference Point This section considers an agent s willingness to pay for insurance. Given that we focus on a single agent, we drop the index i as in the previous section). The following lemma shows that there exists a threshold price such that i) it is a PPE to buy insurance when the price for insurance is below the threshold, and ii) it is a PPE to not buy insurance when the price for insurance is above the threshold. The value of the threshold depends on the distribution of losses F, the reference loss level k that determines the reference point associated with the action of not buying, and the parameters η and λ. Lemma 5. Let p be the solution of the following equation ) EX = p η λ 1 maxp,k) F x)dx k p) +. i) If p < p, then buying is the unique PPE. ii) If p > p, then not buying is the unique PPE. ii) Suppose that p = p. If k p, then buying is a PPE. If k p, then not buying is a PPE. Proof. We first consider the conditions under which different PPEs arise at a given price. We consider three different cases for the relation between reference loss level k and the price p. 15

16 Let First suppose that k < p. Then buying is a PPE if and only if it is a PE by Lemma 3). On the other hand, not buying is a PPE if and only if buying is not a PE because by Proposition 2 a PE always exists). We apply Lemma 2 and find that buying is a PPE if and only if p EX η λ 1 and not buying is a PPE if and only if p EX > η λ 1 p p F x)dx, F x)dx. Now suppose that k > p. Then not buying is a PPE if it is a PE and buying is a PPE if not buying is not a PE these follow from Proposition 2 and Lemma 3). We conclude that buying is a PPE if and only if ) p EX < η λ 1 k F x)dx k p), and not buying is a PPE if and only if ) p EX η λ 1 k F x)dx k p). We finally consider the case that k = p. Similar arguments as in the previous cases show that buying is a PPE if and only if ) p EX η λ 1 k F x)dx k p), and not buying is a PPE if and only if ) p EX η λ 1 k F x)dx k p). Note that the only reason we consider this as a special case instead of including it in one of previous cases) is that both inequalities are weak whereas in the previous cases of inequality is weak and one strict). ) gk, p) p η λ 1 maxp,k) F x)dx k p) +. It follows from the analysis above that i) if gk, p) < EX then buying is the unique PPE, and ii) if gk, p) > EX the not buying is the unique PPE. 16

17 Observe that p has been defined so that gk, p) = EX. We now show that g is strictly increasing in p. In particular, if x 1 < x 2, then gk, x 2 ) gk, x 1 ) = x 2 x 1 η λ 1 maxx2,k) maxx 1,k) F y)dy + η λ 1 k x2 ) + k x 1 ) +) k x2 ) + k x 1 ) +) > x 2 x 1 η λ 1 maxx 2, k) maxx 1, k)) + η λ 1 = 1 η λ 1 ) x 2 x 1 ) >. The monotonicity of g in p implies that if p < p, then buying is the unique PPE; and if p > p, then not buying is the unique PPE. On the other hand, if p = p, then the PPE depends on the reference loss level k: if k p, then buying is a PPE; if k p, then not buying is a PPE. This concludes the proof. We note that the only case that a PPE is not unique is when p = p = k since then both buying and not buying are PPEs). However, we use the convention that if both are PPEs, then the agent buys. On the other hand, if p = p k, then the PPE is unique. Lemma 5 shows that if the price p is smaller than p, then buying is the unique PPE; whereas if the price p is larger than p, then not buying is the unique PPE. We can thus interpret p as the agent s willingness to pay for the insurance product in terms of PPE). 8 The following lemma characterizes the agent s willingness to pay as a function of the reference loss level k and the losses X. With a slight abuse of notation, we write pk, X) to denote that the willingness to pay depends on k and X. 9 It is shown that the agent is willing to pay less for insurance when k is larger. Lemma 6. For given η and λ, let k X) be the solution of Then, k η λ 1 k F x)dx = EX. pk, F ) = { k if k k EX η λ 1 ηλ+1 k )) k F x)dx if k > k 1 1 η λ 1 ηλ+1 8 To be rigorous, if k > p, then p is the supremum of the amounts that the agent would be willing to pay for insurance in terms of PPE. The willingness to pay is normally defined as the maximum amount that a person is willing to pay; however, in this case the agent is willing to pay any price that is strictly smaller than p, so the maximum does not exist. 9 We note that the willingness to pay also depends on η and λ. 17

18 Proof. As in the proof of Lemma 5, let ) gk, p) p η λ 1 maxp,k) F x)dx k p) +. Then, p i k) satisfies gk, p i k)) = EX. We observe that gk, ) < EX and lim p gk, p) =. Also, g is continuous and strictly increasing in p, which implies that gk, p) = EX has a unique solution.) Because of the way that k is defined, we have that gk, k ) = EX, which means that pk ) = k. We observe that if k < k, then k k ) + =, which implies that gk, k ) = gk, k ) = EX. Thus, pk) = k for all k k. We now consider the case that k > k. In this region, gk, k) = k λ 1 k F x)dx, which is strictly increasing in k. Thus, gk, k) > gk, k ) = EX. Since g is increasing in p, we have that gk, p) > EX for all p k. We conclude that pk) < k for all k > k. We now use the fact that pk) < k to simplify the equation gk, pk)) = EX and get 1 η λ 1 ) pk) η λ 1 Solving for pk), we find that in this case 1 pk) = 1 η λ 1 ηλ+1 EX η λ 1 k This concludes the proof. ) k F x)dx k = EX. k F x)dx The following corollary of Lemma 6 provides monotonicity properties of the willingness to pay for insurance with respect to the reference loss level and the distribution of losses. Corollary 1. i) For a fixed X, pk, X) is nonincreasing in k. In particular, pk, X) is constant for k < k X) and strictly decreasing for k > k X). ii) Let X A and X B be two non-negative random variables with finite support. If X A stochastically dominates X B, then pk, X A ) pk, X B ). Proof. i) is a direct consequence of Lemma 6. To show ii), let F A and F B be the distribution functions of X A and X B respectively. Since X A stochastically dominates X B, we have that F A x) F B x) for all x. Our proof is based on the following inequality: EX A + η λ 1 k F A x)dx EX B + η λ 1 18 k )). F B x)dx. 8)

19 To see why this holds, observe that EX A = c c F Ax)dx for any constant c that is greater than maximum element in the support of X A. Thus, EX A + η λ 1 k F A x)dx = c 1 η λ 1 ) k F A x)dx c k F A x)dx for sufficiently large c, which together with the fact that X A stochastically dominates X B implies 8). Lemma 6 and 8) imply that if k maxk X A ), k X B )) then pk, X A ) pk, X B ). To conclude the proof it suffices to show that k X A ) k X B ). Let k A = k X A ) and k B = k X B ) so that k A η λ 1 ka F A x)dx = EX A ; k B η λ 1 kb F B x)dx = EX B. For the sake of contradiction, suppose that k A < k B. Then, EX A EX B = k A η λ 1 ) ka F A x)dx k B η λ 1 < k B η λ 1 = η λ 1 kb EX A EX B, kb F A x)dx ) F B x) F A x))dx k B η λ 1 kb kb F B x)dx F B x)dx which is a contradiction. We note that the first inequality follows from the assumption that k A < k B and the fact that η λ 1 ηλ+1 < 1, the second from 8). Corollary 1 shows that an agent is willing to pay less for insurance when the reference loss level k is larger, or equivalently, when his reference point w k for not buying insurance is smaller. Indeed, an insurance product that provides a hedge against losses becomes less appealing when the reference point is small, because the cases in which the insurance allows the agent not to fall below his reference point and suffer the extra disutility connected with the loss feeling) are more limited. Corollary 1 also shows that the willingess to pay is increasing in the distribution of losses. In particular, for a fixed reference loss level, the agent is willing to pay more for insurance when the distribution of losses is larger in terms of stochastic dominance). This is an intuitive result in the spirit of adverse selection. Corollary 1 provides both the effect of the reference loss level and the loss distribution on the willingness to pay, but considers each effect in isolation. However, in general the reference loss level will depend on the loss distribution. ) ) 19

20 In particular, we expect the reference loss level to be increasing in the distribution of losses. Then, a larger distribution of losses has two effects. On one hand, it implies a larger reference loss level which decreases the willingness to pay according to Corollary 1 i)). On the other hand, the direct effect of a larger loss distribution is to increase the willingness to pay for insurance per Corollary 1 ii)). The interaction of these two effects determines whether the willingness to pay increases or decreases. A necessary condition for advantageous selection is that reference effects dominates so that the bad type is willing to pay less for insurance. We give conditions for advantageous selection in the following section. Straightforward calculations show that k > EX. This implies that if k = EX, then p = k. The following corollary shows that if the reference loss level k is equal to the maximum loss that may occur, then the agent s willingness to pay is equal to the expected value of losses. In other words, if the agent is extremely pessimistic and has the smallest possible reference point, then he is effectively acting as if he were risk neutral. The reason is that in this case the consumption level w X is always greater than the reference point for not buying insurance. The monotonicity properties of pk) given in Lemma 6) then imply that the agent s willingness to pay is strictly greater than EX whenever k is strictly smaller than the maximum value in the support of X. This implies that for any fixed η, λ and a random distribution of losses X, an agent s willingness to pay for insurance is in the interval [EX, k ]. Corollary 2. Let m be the maximum value in the support of X. Then Proof. We first observe that k pm) = EX. F x)dx = k F k) k xdf x). Since m is the maximum value in the support of X, we have that m F x)dx = m EX. This implies that the k of Lemma 6 is smaller than m. Then, by Lemma 6, we have that 1 pm) = 1 η λ 1 EX η λ 1 m )) m F x)dx ηλ+1 = 1 1 η λ 1 ηλ+1 = EX. EX η λ 1 EX ) 2

21 Lastly, we observe that the willingness to pay p is increasing in both λ and η. Recall that λ is the loss aversion parameter; as λ increases, the agents become more loss averse. As a result, both agents value insurance more when λ is larger. On the other hand, η shows how much agents weigh the gain-loss term in their utilities. The effect of an increase is similar: both agents value insurance more. We next turn to conditions under which there exists a unique PE. The following lemma characterizes two prices p and ˆp with p < ˆp) such that buying is the unique PE if the price is smaller than p and not buying is the unique PE if the price exceeds ˆp. Lemma 7. Let p be the solution of ) EX = p η λ 1 k F x)dx k p) +, and ˆp be the solution of EX = p η λ 1 p i) If p < p, then buying is the unique PE. F x)dx. ii) If p > ˆp, then not buying is the unique PE. iii) If p p, ˆp), then both buying and not buying are PE. Proof. Given that there always exists a PE by Proposition 2), buying is the unique PE for agent i if and only if not buying is not a PE, that is by Lemma 2), if ) p EX i < η λ 1 k1 F i x)dx k i p) +. Similarly, not buying is the unique PE for agent i if and only if buying is not a PE, that is by Lemma 2), if and Then, the fact that both p EX i > η λ 1 p η λ 1 p F i x)dx. ) ki F i x)dx k i p) + p η λ 1 p F i x)dx are increasing in p implies that it is a unique PE to buy if p < p i and a unique PE to not buy if p > ˆp. By Proposition 2, p < ˆp. 21

22 We could interpret p as the agent s willingness to pay in terms of unique) PE. However, we note that even though buying is a unique PE when p < p, the agent s action is not well-defined when the price is between p and ˆp. The fact that a unique PE may not exist is the main issue with referring to p as a willingness to pay. It is more accurate to say that p is the price at which buying stops being the unique PE and ˆp is the price at which not buying starts being the unique PE. Similarly to Lemma 6, we can characterize the dependence of p on the reference loss level k. We again slightly abuse the notation and write pk). We show that pk) is increasing for small values of k and decreasing for large values of k. On the other hand, ˆp does not depend on k. Lemma 8. Let k be the solution of k η λ 1 k Then, ˆp = k and EX + η λ 1 k ηλ+1 pk) = F x)dx EX η λ 1 ηλ η λ 1 ηλ+1 F x)dx = EX. k k F x)dx )) if k k if k > k Thus, pk) is increasing for k < k and decreasing for k > k. Proof. We first observe that ˆp is defined as a solution to an equation that does not depend on k. Moreover, k solves this equation. Thus, ˆp = k. Let ) gk, p) p η λ 1 k F i x)dx k p) +. Then, by Proposition 4, p i k) satisfies gk, p i k)) = EX. We observe that gk, ) < EX and lim p gk, p) =. Also, g is continuous and strictly increasing in p, which implies that gk, p) = EX has a unique solution.) Because of the way that k is defined, we have that gk, k ) = EX, which means that pk ) = k. We now observe that gk, k) = k λ 1 k F x)dx, which is strictly increasing in k. We consider the following cases If k > k, then gk, k) > EX. Since g is increasing in p, we have that gk, p) > EX for all p k. Therefore, pk) < k, which implies that 1 pk) = 1 η λ 1 EX η λ 1 )) k k F x)dx. ηλ+1 22

23 If k < k, then gk, k) < EX. Since g is decreasing in p, we have that gk, p) < EX for all p k. Therefore, pk) > k, which implies that pk) = EX + η λ 1 k F x)dx. We note that the k of Lemma 8 is the same as the k of Lemma 6. The following corollary shows that if k is either equal to zero or equal to the maximum loss that may occur, then p is equal to EX. Corollary 3. Let m be the maximum value in the support of X. Then p) = pm) = EX. Proof. By substituting p = in the function that was derived in Lemma 8, we find that p) = EX. The proof for k = m is the same as in the proof of Lemma 2. In contrast to p, the threshold p is decreasing in k when k < k. Recall that the price p is defined as the lowest price at which not buying insurance is a PE. In other words, given the expectation of not buying insurance, p is such that the utility from not buying is equal to the utility from buying. A change in k affects the gain-loss term of both these utilities, leaving the consumption term unchanged. When k increases, both utilities increase. This happens because of two factors: first, the number of contingencies in which the agent perceives a loss weakly) decrease; second, whenever a loss is perceived, the disutility associated to the loss feeling is scaled down. Notice that these two factors are in competition: the more important the first factor is, the less important the second factor becomes; and vice versa. The utilities of buying and of not buying increase at different rates as k increases). Whenever k < p, the utility of buying increases more; the opposite happens when k > p. This is due to the fact that, whenever k < p, the action of buying is always associated with a loss feeling, and the second factor i.e., the scaling down of the loss feeling disutility) dominates. Let us consider a setting with k < k ; at the price level p = pk), not buying is a PE. Notice that k < pk) since k < k ). Given a marginal increase from k to k so that k < p), the gain-loss term of the utility of buying increases more i.e., k µk p) = λ) than the gain-loss term of the utility of not buying i.e., k E[µk X)] < λ). As a result, not buying is no longer a PE: a deviation from not buying to buying is profitable. In order to reestablish not buying as a PE, the price p has to increase to some p. In particular, the consumption term of the utility of buying has to decrease in order to cancel out the relative advantage on the utility of not buying coming from the gain-loss term. Ergo, p = p k ) > pk). Notice that we may expect an agent to buy insurance, even when buying is not the unique PE. This implies that the seller may not price insurance at p. 23