Risk lovers also love insurance

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2 Risk lovers also love insurance Anne Corcos CURAPP-ESS UMR 7319, CNRS and Université de Picardie Jules Verne François Pannequin CES and ENS Cachan Claude Montmarquette CIRANO and Université de Montréal Abstract: This contribution investigates the role played by risk attitudes on insurance demand and challenges the preconception that insurance and risk aversion are exclusively linked. We provide a comprehensive laboratory test of an expanded theory of insurance demand. We resort to a two-part insurance tariff and we adapt the Holt and Laury (2002) s measure of risk aversion to the loss domain a proper measure in the context of insurance to distinguish between riskloving and risk-averse agents. Both experimentally and theoretically, risk attitudes determine insurance demand behavior along with a unit price of insurance and a fixed cost. Empirically, insurance decisions appear to be highly structured around full insurance (36 % of decisions) or no insurance (21 %), and almost 43% of the subjects are risk-lovers. We disentangled the global insurance demand (all individuals) into the propensity to buy insurance and the conditional demand (policyholders only). Other things being equal, (R1) the global demand for insurance is consistently higher for risk-lovers; (R2) when the unit price rises, the global insurance demand shrinks for both types of individuals. This increase has no significant impact on the conditional demand for insurance: those who remain insured do not significantly reduce their coverage. A rise in the unit price reduces only the propensity to buy insurance: risk-lovers, then risk-averters give up. (R3) The impact of the fixed cost is also risk attitude-dependent: when the unit price is above-actuarial, only risk-loving subjects seem to be fixed cost-sensitive, exiting the market as soon as the fixed cost is positive. Keyword: demand for insurance, experimental study, risk-loving and risk-averse participants. JEL classification: C91, D81 We are grateful to Nathalie Etchart-Vincent, John Morgan, Farid Toubal, as well as participants at the Bay Area Behavioral and Experimental Economics Workshop (BABEEW 2013) for their comments and suggestions. Remaining errors or omissions are ours. 1

3 1 Introduction The traditional risk-averse EU model of insurance demand ignores the nonnegligible proportion of individuals who exhibit risk-loving preferences: according to the standard Insurance Theory, risk-loving behaviour and insurance demand are irreconcilable. Addressing the issue of risk-lovers insurance behavior appears therefore irrelevant. However, several insights, both theoretical and empirical, are challenging this point of view. An increasing number of studies and theoretical articles have recently emphasized the fact that risk-loving in the loss domain is quite frequent and should be taken into account. Moreover, Crainich et al. (2013) argue that we have much to learn in investigating the behaviour of risk lovers. They demonstrate that both risk-lovers and risk averse agents exhibit prudence and are inclined to accumulate precautionary savings. In the same way, Kahneman and Tversky s so-called reflection effect (1979) stresses the fact that risk-aversion in the gain domain cohabits with risk-loving in the loss domain which is the field of insurance. To our knowledge, this fundamental finding has never been applied to the understanding of insurance choices. In previous studies, when risk aversion was measured, it was exclusively in the gain domain. Recently, Ottaviani and Vandone (2011) underline the fact that participation in the insurance market could be compatible with a risk seeking behaviour. Their study draws on a sample of 445 households that were subjected to an experimental task involving participation to an Iowa Gambling Task coupled with skin conductance responses to measure their attitude toward, and perception of, risk. The authors confirm that demand of health insurance can be explained by both risk seeking behaviors and the ability of households to anticipate risk. Therefore, the interplay between risk-loving and insurance deserves to be scrutinized. A large number of public policies and regulation rules could be potentially improved by a better knowledge of insurance choices made by risk-lovers. Under what circumstances both theoretical and experimental are they liable to buy insurance or reject all insurance coverage? How does the insurance behaviour of risk-loving individuals differ from that of risk-averse individuals? It is surprising that no theoretical model or experimental study have examined the role of risk attitudes in the demand for insurance in a simplified context, although several experiments have addressed the issue of the supply and demand insurance behaviour in the presence of asymmetric information (see for example, Shapira and Venezia, (1999)), and have investigated the experimental equilibrium in an insurance market with asymmetric information (Goswami, Grace, and Rebello, 2008), or in competitive insurance markets (Riahi, Levy-Garboua, Montmarquette, 2013). 2

4 While real data do shed some light on the demand for insurance, their main weakness resides in the treatment of risk aversion, due to the difficulty of measuring it. For lack of a better option, econometric studies either completely ignore the role played by attitudes toward risk or have to rely on proxy variables. For example, Esho et al. (2004) confirmed the negative impact of risk aversion on the demand for insurance. Their result is based on the assumption that education and risk aversion were positively correlated: according to this approach, risk aversion increases with education. Since education allows individuals to acquire a better understanding of risk, the choice of education as a proxy variable is legitimated. The authors also used a cultural indicator derived from survey data, the uncertainty avoidance index, which appeared to go hand in hand with a higher level of coverage. On the experimental side, original laboratory studies on insurance demand have more in common with survey-based studies than will experiments. For example, Slovic et al. (1977), Schoemaker and Kunreuther (1979), and Hershey and Schoemaker (1980) used questionnaires to demonstrate how the risk aversion-based expected utility criteria is poorly suited to explain insurance behaviour when individuals are more inclined to be risk-loving than risk-averse in the domain of losses. These studies report that individuals frequently refused to buy full insurance even when they are offered actuarial rates. Hershey and Schoemaker (1980) also mentioned the presence of a context effect, leading individuals to display more risk averse decisions when the notion of insurance is explicitly evoked. A resurgence of interest in the experimental approach of insurance demand has occurred in the recent years. Kunreuther and Michel-Kerjan (2012) took an experimental approach to test the advantages of multi-year property insurance over annual contracts. They found, amongst other things, that the probability of purchasing insurance increases with risk-aversion. In the previous studies, if authors controlled for risk aversion, they were only eliciting it in the gain domain. As a consequence, risk loving attitude was not really considered. Instead, it was understood as a weak degree of risk aversion. However, aside from the fact that they do not include either controls for risk attitude or monetary incentives, previous experimental studies are also characterized by the simplicity of contractual insurance mechanisms: the unit price of insurance is always actuarial, and the only granted insurance level is the complete coverage. Thus, the individual has to choose between full insurance and no insurance. But difference between risk loving and risk aversion is not only a matter of unit price sensitivity: while insurance demand of a risk-averse individual continuously changes with the unit price, a risk-loving individual is only 3

5 attracted by two utmost solutions: full risk retention or full insurance coverage (if over-insurance is forbidden). Our paper takes a theoretical and an experimental look at the sensitivity of insurance demand to premiums, accounting for the role of attitudes toward risk. The theoretical model we present generalizes Mossin's (1968) canonical insurance demand model in two ways: by extending the analysis to include risk-averse, risk-neutral, and risk-loving individuals and by mobilizing a more general insurance pricing scheme a two-part premium structure that includes a fixed and a variable component the fixed cost and the unit price, to better fit the real world. Clearly, insurance appears determined by a large variety of fixed administrative costs that need to be paid regardless of the level of coverage; moreover, the level of coverage is associated with a unit price that varies with the amount of insurance purchased. In many countries, health insurance involves a mandatory contribution that is complemented by a user fee. For the insured, this design amounts to paying a fixed cost that is then adjusted to reflect the quantity of medical services consumed. This pricing scheme makes it possible for an expected utility maximizer-agent to choose, not only whether to buy insurance, but also how much insurance she wants. Our experiment has been designed to test the predictions of our theoretical model depending on risk attitudes and contract tools. The Holt and Laury adapted protocol classifies individuals depending on their attitude toward risk (loving or averse) in the domain of losses. Both theoretical and experimental results emphasize the very difference between risk lover and risk averter s insurance demand: the impact of an increase in the unit price or the fixed cost is risk-attitude dependent. It operates mostly through propensity to purchase insurance, and, to a lesser extent, through the conditional demand. In the same way, our model and our experiment provide evidence that insurance and risk-loving are not incompatible. Distinguishing risk-loving and risk-averse individuals from both theoretical and experimental point of view and trying to characterize their insurancerelated behaviour when confronted to two fundamental pricing tools of the insurance premium, leads to a deeper comprehension of insurance demand. The remaining of the paper is organized as follows. Section 2 presents the theoretical model of insurance demand in the presence of a two-part premium structure. It emphasizes the consequences of attitudes toward risk - risk-averse, risk-neutral, or risk-loving - and yields a series of predictions that can be tested experimentally. Section 3 describes the two stages of the experimental design: measuring attitude toward risk and eliciting the insurance demand at the individual level. Section 4 presents the experimental results and shows in what fashion both the contractual parameters (unit price and fixed cost) and attitude towards risk shape the insurance demand 4

6 behaviour. Section 5 presents our two stage econometrics model of insurance demand and section 6 concludes with a discussion regarding the consequences of our experimental findings, both in terms of the validity of the theoretical predictions and the contractual policy. 2 The Theory of Insurance Demand The theoretical model we present here generalizes Mossin's (1968) canonical insurance demand model in two ways: On the one hand, by extending the analysis to include risk-averse, risk-neutral, and risk-loving individuals, on the other hand, by introducing a more general insurance pricing scheme that includes a fixed and a variable component. 2.1 Insurance demand with a two-part premium structure Consider an expected-utility-maximizer agent with an original level of wealth W 0, subject to a risk of losing some amount x with probability q. This individual has the option of acquiring a guaranteed compensation amounting to I in the event of an accident if he or she pays an insurance premium P = pi + C, where p represents the unit price of insurance, I the indemnity, and C a fixed cost. We subsequently assume that 0 I x, thus precluding over-insurance. Final wealth, where W 1 is the wealth in case of accident and W 2 the wealth in case an accident occurs, is given by: W 1 = W 0 pi C W 2 = W 0 pi C x + I We examine the insurance decisions of this agent as a function of her attitude towards risk. Her preferences are represented by a concave or convex utility function U(W) and she maximizes the following expected utility: EU(I) = (1 q)u(w 1 ) + q U(W 2 ) = (1 q) U(W 0 pi C) + q U(W 0 pi C x + I) The theoretical model provides the necessary conditions for this agent to buy insurance and, if she does, the optimal level of insurance coverage. Agent will therefore opt for insurance if his welfare with an insurance contract is greater than his welfare without, as expressed by the following participation condition (PC), where EU 0 = (1 q) U(W 0 ) + q U(W 0 x) represents expected utility in the absence of insurance: 5

7 EU(I) EU 0 (1 q)u(w 0 pi C) + q U(W 0 pi C x I) EU 0 (PC) We examine the optimal insurance demands depending on the agent s risk attitude: risk-aversion, risk-neutral, or risk-loving For a risk-averse agent, if condition (PC) is satisfied, the optimal level of coverage is characterized by the following first-order condition: EU = p(1 q)u (W I 1 ) + (1 p)qu (W 2 ) (1) Under the assumption of risk aversion, marginal utility is decreasing and the following second-order condition is satisfied: 2 EU I 2 = p2 (1 q)u (W 1 ) + (1 p) 2 qu (W 2 ) < 0 (2) When the unit price of insurance is actuarial (p = q), it is optimal for the individual to buy complete coverage (I 0 = x) and individual welfare is given by: EU = U(W 0 qx C) 1. However, this utility is decreasing in C, so there is a threshold C 0 beyond which the individual prefers to remain uninsured. Thus, if: C C 0, I 0 = x C > C 0, I 0 = 0 the threshold C 0 represents the fixed cost at which the individual is indifferent between full actuarial insurance and no insurance: U(W 0 qx C 0 ) = EU 0. For an above-actuarial unit price of insurance (p>q), individual opts for partial insurance coverage 2 and the highest level of welfare is lower in the absence of a fixed cost than in the case of an actuarial price 3. Therefore, the fixed cost C 1 beyond which the individual forgoes any insurance is lower than C 0. Given I 1, this follows from the following condition: 1 Indeed, condition (1) implies that U (W 1) = U (W 2), which leads to I 0 = x. 2 Since p > q, the f.o.c. implies that U (W 1) < U (W 2). It follows that W 1 > W 2 and I < x. 3 For an above-actuarial price of insurance p (p>q), welfare is maximized at the optimal level of coverage I 1. For this same coverage I 1, the welfare would be higher if the price of insurance was q, but at this price full insurance would be optimal. The following inequalities summarize this reasoning: (1-q)U(W 0-pI 1 ) + qu(w 0-pI 1 -x+i 1 ) < (1-q)U(W 0-qI 1 ) + qu(w 0-qI 1 -x+i 1 ) < U(W 0-qx). 6

8 (1 q) U(W 0 pi 1 C 1 ) + q U(W 0 pi 1 C 1 x + I 1 ) = EU 0 For a below-actuarial unit price of insurance (p<q), the highest level of welfare is greater in the absence of a fixed cost than in the other two cases. If they are given the opportunity, individuals will choose to over insure 4 and the highest fixed cost they will be able to assume (C 2 C 0 ) is characterized by the following condition (where I 2 represents their optimal level of coverage): (1 q) U(W 0 pi 2 C 2 ) + q U(W 0 pi 2 C 2 x I 2 ) = EU 0. In other words, when the unit price is lower or equal to its actuarial level, risk-averse individuals' insurance demand is binary: depending on the level of the fixed cost, they will either buy complete insurance or buy no insurance at all if the fixed cost of insurance is considered as prohibitive. When the unit price is higher than actuarial, the insurance level becomes continuous but risk averters do no longer buy complete insurance. Furthermore, since the certainty equivalent of a risky situation decreases with risk aversion, the threshold C 0 increases with individuals' risk aversion The optimal insurance demand of a risk-neutral agent is as follows: For an actuarial price (p = q), the individual is indifferent to the level of coverage (I 0 [0,x]) if C = 0 and chooses no insurance if C > 0). For an above-actuarial price (p > q), no insurance is purchased (I 1 =0) at any fixed cost (C 0). For a below-actuarial price (p < q), given the constraint on I (0 I x), agents' demand for insurance is dichotomous again: they choose full insurance (I 2 = x) when the fixed cost is nil; if C > 0, they either buy full coverage (I 2 =x) or do not get insurance if the amount of the fixed cost is dissuasive (I 2 = 0) For a risk-loving agent, expected utility is a convex function of the benefit l. If marginal utility is increasing (U (W) > 0) the second order conditions are positive and only corner solutions (no insurance and full coverage) are likely to be observed. For an actuarial or above-actuarial unit price of insurance (p q), first order condition (f.o.c.) (1), evaluated at the point of no insurance (I=0) is negative; this is also true when evaluated at the point of full insurance (I=x), as shown by the following equations 5 : 4 Since p < q, the f.o.c. implies that U (W 1) > U (W 2). Consequently, I > x. 5 Since W 0 > W 0-x, U (W 0) > U(W 0-x), because marginal utility is increasing for a risk-loving individual. 7

9 EU I EU I I = 0 I= x = p(1 q) U '( W ) + (1 p) qu ( W x) < 0 = ( q pu ) ( W px) In other words, due to the convexity of expected utility, the decreasing segment of the function EU(I) is the geometrical locus of all insurance coverages (for 0 I x). In this case, the optimal demand for insurance is zero. For a below-actuarial price of insurance (p < q), risk-loving individuals will choose to either self-insure (I 2 = 0) or buy full insurance (I 2 = x) if the unit price is sufficiently low. In fact, in this case the minimum of the function EU(I) is on the left of the point of full insurance and it is very plausible that full insurance will be preferred to assuming the risk (EU(x) > EU(0)), as shown in the graph below. (3) EU(I) EU(x) EU(0) 0 I=x I Figure 1: Expected utility of a risk-loving individual as a function of their insurance demand (case of a below-actuarial price) 8

10 Consequently, a risk-loving agent will always take a binary decision: buying full insurance only if its unit price is sufficiently below the actuarial price buying no insurance otherwise. As above, since her welfare declines in the presence of a fixed cost, it is possible to determine for any insurance unit price p < q that is sufficiently attractive to make full insurance preferable to no insurance the threshold cost C 2 that leaves a given individual indifferent between these two corner solutions. 3 Experimental Design 3.1 The tested hypothesis The experiment has been designed to test empirically the validity of our theoretical model of insurance demand. The use of a two-part tariff combining 3 unit prices below-actuarial, actuarial and above-actuarial price and 2 fixed cost levels makes it possible to elicitate insurance demand according to agents attitude towards risk. Table 1 below summarizes the theoretical predictions for insurance demand in compliance with our previous theoretical model. Table 1: Insurance demand depending on contractual features and attitude towards risk Below-actuarial price p < q Actuarial price p = q Above-actuarial price p > q C = 0 C > 0 C = 0 C > 0 C = 0 C > 0 Risk I = x I I = x I averse {0,x} {0,x} I [0,x[ I [0,x[ Risk I = x I I [0,x] I = 0 I = 0 I = 0 neutral Risk loving I {0,x} {0,x} I {0,x} I = 0 I = 0 I = 0 I = 0 Several features clearly stand out from the table. Regardless attitude towards risk, the demand for insurance tends to decrease as the price (or the cost) increases. Regardless the premium scheme, the demand for insurance tends to decrease as risk aversion falls. Our experiment can therefore provide not only the analysis of risk attitudes effects on insurance demand but also demand elasticity to the premium components and a goodness of fit test of observed demands to theoretical demands. 9

11 1) Quality of the matching of observed demand with theoretical demand The predictive power of the model, i.e. the goodness of the fit between observed demand and its theoretical counterpart, will depend on the effects of the three key variables identified: the fixed cost (C), the unit price of insurance (p), and the attitude toward risk. To the extent that individual is not allowed to over insure (in our experiment, I x), she will most often trade-off between full insurance (I = x) and no insurance (I = 0) if the fixed cost (0 or C) is considered as prohibitive. Situations of partial insurance (0 < I < x) will only arise if the unit price is higher than actuarial (p > q) and the person is risk averse. 2) Impact of a variation in p on the demand for insurance As Eeckhoudt, Gollier and Schlesinger (2005) demonstrate, if insurance is an inferior good, the premium-effect will be ambiguous: an increase in the insurance premium will have a positive income effect and a negative substitution effect so the final effect is undetermined. Conversely, if insurance is a normal good, these two effects will reinforce each other and an increase in the unit price, p, will reduce the demand for insurance. 3) Variation in C and retention of risk An increase in fixed cost may completely deter the demand for insurance. This is all the more true as risk aversion is weak. 4) The role of risk attitude Demand for insurance increases with risk attitude. The gap between the demand of risk-averse and risk-loving agents gets wider in the presence of high insurance unit prices. Thus, for a given unit price of insurance, the highest acceptable fixed cost (whether positive or zero), increases with the degree of risk aversion. In other words, for a given fixed cost, the likelihood for a risk-loving individual to choose not to buy insurance is greater. 3.2 The practical modalities The experiment was conducted in Montreal with 117 participants, both male and female. The subjects were students and workers of various ages. The subjects were seated in front of a computer and, confronted with several situations involving risk, for each risky situation they had to decide whether to buy insurance, and how much. Since the subjects had to make insurance decisions to deal with the risk of loss, a first step involved the elicitation of their attitude toward risk in a domain of losses. The second step addresses the insurance demand issue. 10

12 Step 1: Measuring the attitude toward risk in the domain of losses We adapted Laury and Holt s method (2002) for eliciting risk aversion from (multiple price list) to the loss domain, following Chakravarty and Roy s work (2009). The subjects had to make 10 decisions (one decision = one line), each decision consisting in choosing between two risky lotteries generating losses. Table 2 describes the 10 successive decisions. Table 2: Measurement of risk attitudes Decision Option A Option B % likelihood Loss (in $) % likelihood Loss (in $) % likelihood Loss (in $) % likelihood Loss (in $) Expected Payoff Difference E(A)-E(B) This procedure allows each subject s degree of relative risk aversion to be estimated in the form of an interval. Risk-aversion is determined by the number of times in which option A the least risky one is chosen by the subject. In Table 2, the point at which individuals switch from option A to option B reveals their level of risk aversion 6 : they are risk loving if they switch (strictly) before question 5, and risk neutral or risk averse beyond that point. In our protocol, risk neutral individuals are indifferent between option A and option B at question 5 and prefer option A for the first four decisions and option B for the last five. Consequently, perfectly risk-neutral individuals will never switch before question 5; however, they may only 6 Following Chakravarty and Roy (2009), we could assume that the subjects utility functions are CRRA (Constant Relative Risk Aversion) i.e. such that u(w) = - (-w) k with w < 0. Then, by observing when a given subject switches from option A to option B, it is possible to identify into which interval her relative risk aversion falls. In the present study, we don t need to calculate these intervals since we are only interested in discriminating between risk lovers and risk averters. 11

13 switch at question 6. For this reason, risk neutral individuals are grouped together with those who are risk averse. It is worthwhile to notice that this measure of risk attitudes is only used to separate risk lovers from risk averters. Thus, our method is not depending on the utility function since we are not focusing on the intensity of risk attitudes but on their nature. The amount of losses sustained (on the order of $5) was comparable to the amounts at stake later in the insurance decisions Step 2: Eliciting the demand for insurance The second step of the experiment consists in a series of 6 rounds of insurance decisions. At the beginning of each period, the subjects were granted an endowment W 0 = 1000 EMU (Experimental Monetary Units). At each period they ran a 10 % chance of having an accident that would cost them the entire 1000 EMU. They had the option, but were not obliged, to buy an insurance against that risk of loss: subject to the payment of a premium P, which was due at the beginning of the period, subjects received an indemnity I if an accident occurred during the period 7. The premium increased with the desired level of the indemnity. The various premiums and indemnities were determined according to a twopart insurance premium: P = pi + C where P = insurance premium, p = unit price of insurance; I = quantity of insurance or indemnity, C = fixed cost. A unit price p = 0.1 corresponded to the actuarial price. Thus, after the decision of whether or not to buy insurance had been made, an individual random draw determined whether an accident had occurred. At the end of each period determined whether an accident had occurred or not. The computer then calculated the subject s final wealth and posted it on screen. Her wealth amounted to W 1 or W 2 depending on whether she suffered a loss. W 1 = W 0 P = W 0 pi C W 2 = W 0 P x + I = W 0 pi C x + I This decision period was then replayed 5 more rounds, with each of the five other premium schedules obtained by crossing three unit price levels (the actuarial price p = 0.1, a below-actuarial price p = 0.05, and an aboveactuarial price p = 0.15) and two levels of fixed cost (C = 0 or C = 50 EMU). Individuals were confronted with these schedules in a random sequence so as to control for any potential order effect. Each round (or period) was independent: the subjects received a new endowment of 1000 EMU regardless of their gains or losses in the previous round. 7 With a probability of 10 %. 12

14 Table 3 shows the premium grid the correspondence between the proposed premiums and benefits computed for the actuarial unit price p = 0.1 and a fixed cost C = 0. For example, it would cost subject a premium P = 70 EMU at the beginning of the period to receive compensation I = 700 EMU in case an accident occurs during the period. At the end of the period, her wealth would be W 1 = = 930 if no accident occurs and W 2 = = 630 if an accident occurs. Premium = Total cost of insurance p = 0.1 Table 3: Insurance premium grid Indemnity: Reimbursement in case of accident Wealth at end of period If no accident If accident C = premium premium indemnity 13

15 Clearly, if the subject chooses not to buy insurance, her premiums and indemnities would be nil (see 1st row of Table 3). At the end of the period, her wealth would be 1000 if no accident occured and 0 if some accident occurred Monetary Incentives Subjects' gains had a threefold component determined as follows: one of the six insurance decision periods was drawn at random. The gain from that period was converted into dollars at the rate 1 EMU = 0.5 cents. In order to motivate the lottery choices at that stage, the computer randomly drew one of the subject's 10 decisions, as well as a number between 1 and 10, in order to determine the loss associated with the chosen option. This gain was topped off with a flat $5 bonus for participating in the experiment to compensate for the average loss in the risk aversion measurement of step 1. These draws, as well as the resulting losses, were only communicated at the end of the experiment to prevent intermediate wealth effects from influencing subjects' later decisions. All these rules were fully known to the subjects before the beginning of the experiment. On average, the amount earned represents $15 on an hourly basis. 4 Results Almost 43% of the subjects have been proved to be risk-lovers. The insurance decision appears to be highly structured (57 % of decisions) around the choice of full insurance (36 % of insurance decisions) or no insurance (21 %), cf. Figure 1. For all contracts, the subjects chose an average coverage amounting to 556 EMU, for a mean prime of 71 EMU. 8 8 The demand for insurance, by contract, is given in Table 5. 14

16 P e r c e n t Figure 2: Breakdown of indemnities selected 4.1 The matching of observed demand with theoretical demand For each contract, Table 4 below provides both the theoretical demand for insurance, the observed distribution (as a percentage of the population) of participants for each level of coverage, as well as a test of the goodness of the fit between observed and theoretically expected distributions of demand, depending on whether individuals are risk loving or risk averse (or neutral). In the five cases where the theoretical expectation is a unique value (e.g. I* = x or I* = 0), we rejected H 0 (RH0) in two cases out of five. In each of the four cases where the theoretical expectation takes two values {0 ;x}, H 0 is accepted (AH0), and the observed behaviour appears to be consistent with theoretical predictions. 9 In the two cases where the theoretical demand takes values in the interval [0 ;x[, all the values inferior to 1000 were grouped. We tested for the goodness of fit with a theoretical binary distribution in which the demand for 1000 is nil. In each of these cases, we were led to reject the null hypothesis at the 5% level (two-tailed test). This result suggests that riskaverse individuals have a strong preference for complete coverage, even in the presence of an above-actuarial unit price and a fixed cost. In the last case, the predicted demand for insurance is given by some distribution over the interval [0 ;x]. Any observed distribution will thus be consistent with the theoretical prediction. 9 Where x is the 1000 loss. 15

17 Finally, Table 4 allows us to draw the following conclusions: First, in the presence of a below-actuarial price, the observed demand for insurance of both risk lovers and risk averters is not statistically different from that predicted by theory. The insurance demand of risk-averse individuals is also consistent with the theoretical expectation when the unit price of insurance is equal to the actuarial price. In the same way, when the fixed cost is positive, risk-lovers act in compliance with theoretical model and exit the insurance market when the unit price is at least actuarial. Second, by contrast, in other cases, both risk-lovers and risk-averse individuals seem less price-elastic than predicted by theory: in the presence of a fixed cost of zero, and only in that case, does a high unit cost not discourage risk-lovers complete coverage. In the same way, when the unit price of insurance is high (above the actuarial price), risk-averse individuals exhibit a unit price elasticity lower than predicted by theory: in this case, all participants do not completely reject full insurance. To summarize, in 8 of the 12 cases, observed behaviour is consistent with the theoretical predictions 10, mostly situations where subjects are expected to exit the market (I* = 0) or to buy full insurance (I* = x). In the four cases that deviate from the theoretical pattern, both risk-loving and risk-averse subjects exhibit insufficient elasticity of demand, regarding the unit price insurance for risk-averters and the fixed cost for risk-lovers : risk-lovers choose to buy insurance instead of leaving the market (2 cases out of 4) while risk-averse subjects buy full insurance when they are expected not to do so (2 cases out of 4). 10 More precisely, 66.2 % (resp %) of insurance decisions made by risk-averse agents (resp. risk-loving) are consistent with the theoretical predictions. This corresponds to 57 % of the insurance decisions made by all individuals. 16

18 Table 4: Test for goodness of fit of observed distributions with theoretical distributions Risk aversion (and neutrality) Risk loving Belowactuarial price p < q Actuarial price p = q C = 0 C > 0 C = 0 C > 0 I* = x AH (0.437) I* {0,x} AH (0.641) I* [0,x] AH0 I* {0,x} AH (0.298) I* {0,x} AH (0.972) I* {0,x} AH (0.883) I* = 0 RH ** (0.0041) I* = 0 AH (0.116) Aboveactuarial price p > q C = 0 C > 0 I* [0,x[ RH0 5.39* (0.0203) I* [0,x[ RH0 5.39* (0.0203) I* = 0 RH * (0.0389) I* = 0 AH (1.113) Chi-square and p-value (between parentheses) are reported. Thresholds: *5 %; ** 1 %. 17

19 4.2 The Determinants of insurance demand We next turn our attention to analysing the impact of the unit price and the fixed cost on global demand for insurance distinguishing between its two components: the propensity of individuals to buy insurance on the one hand, and the demand for insurance of those who do (the conditional demand for insurance) 11, on the other hand. Figure below presents the evolution of the global demand for insurance as a function of its unit price. 700 Global Insurance demand Averses (C=0) Lovers (C=0) Averses (C=50) Lovers (C=50) 0,05 0,10 0,15 Averses Lovers Price Figure 3: Evolution of global insurance demand: Table 5 shows the global demand (GD) for insurance (I 0) as well as the conditional demand (CD) for insurance of those who actually buy some insurance (I > 0). The average proportion of individuals who buy insurance (PI) is reported in Table 6. In each table, values are reported by type of contract and risk attitude. 11 Conditional on actually buying insurance. 18

20 Table 5: Insurance demand Global Demand GD Base: N =117 (I 0) Conditional Demand CD Base :N s.t. I > 0 Averse Loving Combined Averse Loving Combined Unit price Combined Fixed cost (N = 67) (N = 50) Tot Tot Tot Tot ( N ): Number of observations (N = 117) (N) (N) (N) (59) (58) (117) (58) (58) (116) 680 (50) (51) (101) (167) (167) (334) (42) (41) (83) 700 (39) 700 (31) 700 (70) (36) (29) (65) (117) (101) (218) (101) (99) 757 (200) (97) (89) (186) (86) (80) 635 (166) (284) (268) (552) Two key facts stand out from Figure as well as from Table 5 and Table 6. First, when aggregating data for all the subjects, whatever the fixed cost, the global demand for insurance appears to shrink as the unit price rises. This effect can operate through a decrease of both the propensity to buy insurance and/or the conditional demand. Second, risk attitude affects both the level of insurance demand and the way people react to a change in fixed cost. o The global demand for insurance of risk-averse individuals appears to be higher than that of risk lovers. 19

21 o As noted above, risk-lovers are fixed cost sensitive whereas risk-averters are not: their demands are almost the same for the two levels of the fixed cost. Unit price 0.05 Table 6: Propensity to buy insurance Base Fixed cost Averse N = 67 Loving N = 50 Combined N = Tot Tot Combined Tot Tot N: Number of observations Increases in unit price insurance (p) and insurance demand The impact of a higher unit price on global insurance demand results from two effects: higher price may have an impact on subject s propensity to buy insurance (in particular by deterring her from insuring), but it may also provide an incentive, for those who do buy insurance, to reduce their coverage (lower conditional demand). 20

22 Table 7: Insurance demand and increases in Unit price Variation in the proportion of insured PI N = 117 (1) (2) (3) Variation in the Conditional insurance demand CD N = those who buy insurance I > 0 Overall effect: Variation in the Global insurance demand GD N = 117 Increase in unit price p Risk attitude Difference PI<0 t-tests (p-value) Difference CD t-tests (p-value) df Difference GD t-tests (p-value) df 0.05 to to 0.15 Risk-averse Risk-loving Combined Risk-averse Risk-loving Combined Thresholds: ** 5 % ; *10 % df = degrees of freedom (0.43) (0.015)** (0.044)** (0.009)** (0.225) (0.016)** (0.576) (0.255) (0.241) (0.124) (0.102) (0.024)** (0.587) (0.019)** (0.043)** (0.006)** (0.092)* (0.0018)** 466 Table 7 reports the results regarding the impact of an increase in the unit price on both propensity to insure (col. 1), conditional and global demands (col. 2 and 3). Each cell provides their algebra variation (1 st row), t-test (2 nd row) and p-values (3 rd row). 4 th row, when exists, provides the degree of freedom of the test. In column 3 we see that an increase in the unit price of insurance has mostly a negative impact on global demand: the level of coverage falls as the unit 21

23 price of insurance rises, regardless the subject s risk attitude. 12 This decrease is primarily driven by the propensity to buy insurance (column 1). The impact on conditional demand of an increase in the unit price (column 2) is never significant except a negative one at the aggregate level (for all individuals) when the unit price exceeds the actuarial price 13. Conversely, as the unit price rises, individuals are sequentially crowded out from the insurance market, risk-lovers first, risk-averse subjects then (column 1) Increase in the fixed cost (C) and the retention of risk An increase in the fixed cost is also likely to modify the global demand for insurance via both the propensity of individuals to buy insurance and the amount of coverage chosen by those who continue to do so. Table 8 shows the impact of these two effects on global insurance demand. The compared analysis of global demand with the matched samples using a t-test (column 3) suggests that an increase in the fixed cost has no significant impact on the global insurance demand. This t conclusion must be qualified by an analysis of the impacts of a fixed cost increase on the propensity to buy insurance and the level of coverage purchased. A one-tailed proportion comparison test (column 1) reveals that an increase in the fixed cost significantly reduces the propensity of risk-loving agents to buy insurance when the unit price is at least actuarial. It leaves the propensity at of risk-averse agents unchanged. Conversely, a t-test reveals that, whatever their risk attitude, individuals who choose to buy insurance do not significantly reduce the amount of coverage following an increase of the fixed cost (column 2). 12 Except for risk-averse individuals when the price remains attractive (i.e. when it moves from below-actuarial to actuarial). 13 Of course, when unit price increases three-fold (from 0.05 to 0.15), a comparison test of conditional demand is significant. This result must be qualified insofar it represents a 200 percent increase! 22

24 Table 8 : Increase in the fixed cost and insurance behavior Unit price P Risk attitude Risk-averse Risk-loving Combined Risk-averse Risk-loving Combined Risk-averse Risk-loving Combined (1) Variation in the proportion of insured PI N = 117 Difference PI<0 t-tests (p-value) df (0.398) (0.395) (0.355) 0 0 (0.5) (0.040)** (0.098)* (0.5795) (0.0711)* (0.194) (2) Variation in the Conditional insurance demand CD N = those who buy insurance I > 0 Difference CD t-tests (p-value) df (0.339) (0.304) (0.951) (0.838) (0.592) (0.596) (0.865) (0.2014) (0.471) 446 (3) Overall effect: Variation in global insurance demand GD N = 117 Difference GD t-tests (p-value) df (0.698) (0.496) (0.77) (0.791) (0.195) (0.24) (0.679) (0.289) (0.29) 23

25 Combined Thresholds: * 10 % ; **5% df = degrees of freedom Risk-averse Risk-loving Combined 0 0 (0.5) (0.019)** (0.070)* (0.535) (0.701) (0.458) (0.876) (0.122) (0.24) These results are highly compliant with theoretical predictions where the only expected effect when it exists of an increase of the fixed cost is to crowd subjects out of the insurance market. Risk-lovers are highly fixed cost sensitive: in compliance with theoretical predictions, instead of decreasing the level of coverage, they prefer to leave the market. On the contrary, the insurance behaviour of risk-averse subjects conditional demand and propensity to buy insurance seems to be fixed cost-inelastic. As for the unit price, risk attitudes influence the way the subjects react to an increase of the fixed cost. 5 The econometric model: two-stage estimation Our econometric analysis of insurance demand fits into our framework. The insurance decision is twofold. First, subject has to decide whether or not he buys insurance (model 4). Second, conditionally on actually buying insurance, she has to choose the level of his insurance coverage (model 5). Cost components (unit price and fixed cost) as well as risk attitudes provide explanatory variables of the models. Finally, a last model examines the determinants of the decision to opt for partial or full insurance conditional on the decision to buy insurance. 5.1 The propensity to buy insurance Since the theoretical model allows us to simultaneously characterize the conditions under which individuals will, or will not, buy insurance, and to explain the level of coverage if they do, we apply a two-step model that distinguishes between the choice to buy insurance and the level of coverage (indemnity). The decision to buy insurance (Y = 1) depends on the characteristics of the insurance contract as well as on the subject s risk- 24

26 aversion (risk-loving) profile in the loss domain. Consequently, we use a probit with panel data; see equation (4). P( Y = 1) = F( α + α DACTP + α DACTM + α DCOST α RiskLover ++ α DMTACT * RiskLover α DACTP * RiskLover + α DCOST 50* RiskLover + ε 6 7 (4) where w z 1 2 F ( w) = e dz 2π 2 RiskLover is an auxiliary variable equal to 1 for risk-loving individuals, and 0 for the risk averse. DACTP, DACTM, and DCOST50 are also auxiliary variables that describe the pricing of the insurance contract: DACTP = 1 if the contract unit price is inferior to the actuarial price; DACTM = 1 if the contract unit price is higher than the actuarial price; and DCOST50 = 1 if the cost of the contract is C = 50. Each of these variables was crossed with the variable for individuals' risk attitude. In compliance with the theoretical model, we expect a below-actuarial price to increase the likelihood that individuals buy insurance (α 1 > 0). Conversely, an above-actuarial unit price is expected to decrease the probability of buying insurance (α 2 < 0). The probability of buying insurance should decrease when the fixed cost increases (α 3 < 0) as well as for risk-lovers (α 4 < 0). The unit price effect should also depend on the risk attitude, risk-lovers being less price sensitive, especially for above-actuarial prices; the maximum fixed cost that is compatible with buying insurance increases with risk aversion. So, for risk-loving agents the probability of buying insurance should decrease with cross unit price effects (α 5 and α 6 < 0), and with the cross effect of the fixed cost (α 7 < 0). 25

27 VARIABLES Table 9: Estimates of insurance demand models (1) (2) (3) Insurance Conditional demand Partial vs. full Indemniy if insurance insured coverage DACTP: 1 if the unit price is belowactuarial (0.05), 0 otherwise * (0.757) (0.385) (0.074) DACTM: 1 if the unit price is aboveactuarial (0.15), 0 otherwise *** * ** (0.003) (0.076) (0.031) DCOST50: 1 if cost=50, 0 otherwise (0.965) (0.996) (0.646) RiskLover: 1 if AP<5, 0 otherwise * (0.075) (0.936) (0.328) DACTP & RiskLover 0.654* (0.078) DACTM & RiskLover (0.125) DCOST50 & RiskLover * (0.059) DCOST50 & DACTP (0.910) (0.844) DCOST50 & DACTM (0.793) (0.342) Inverse of Mills ratio (0.716) (0.618) Constant 1.932*** *** 0.447*** (0.000) (0.000) (0.000) Observations Number of subjects ll_c chi df_m Ll Rho p-value in parentheses: *** p < 0.01, ** p < 0.05, * p < 0.1 Estimation of model (4) using a probit specified on binary panel data, yields results that are consistent with our expectations and validate the theoretical 26

28 results (see Table 9 column (1)). The most significant estimated coefficient is for variable DACTM (above-actuarial price): at this unit price, the probability of buying insurance drops off markedly. At the 10 % confidence level (or 5 % for one-tailed tests), risk-loving individuals buy less insurance than risk-averse ones. Furthermore, it is of interest to note that risk-loving individuals confronted with a below-actuarial price are motivated to buy insurance. While the coefficient of cost is not significant, its influence operates through the fixed cost risk-preference cross effect. 5.2 The insurance demand model To estimate the level of indemnity, we use a linear model (5) on panel data, correcting for selection bias with the panel probit formulation in equation (4). Indemnity = β + β DACTP + β DACTM + β DCOST50 + β Risklover β DACTM * DCOST50 + β DACTP * DCOST50 ++ β IRM + θ (5) As the decision to insure or not, the level of indemnity is also a function of the features of the insurance contracts, subjects' risk aversion, and the costunit price cross variables. However, we now expect the below-actuarial price not to increase the level of indemnity: theory predicts that subjects should opt for full insurance 14. So, β 1 should be non-significant. Conversely, the presence of an above-actuarial price is expected to have a negative impact on the level of indemnity: the higher the unit price, the less interest individuals have in extensive insurance (β 2 < 0). The amount of the coverage may, on the other hand, vary with the presence or absence of a fixed cost if insurance is an inferior good 15 (β 3 > 0). In this case, its impact on the level of coverage will also depend on unit prices: If the unit price is below-actuarial, we would not expect to observe a cost effect on the level of coverage (β 5 not significant); if, on the contrary, the unit price is above-actuarial, the level of indemnity should increase due to the effect of the fixed cost (β 6 significant and positive). Furthermore, in the case of constant risk aversion coefficients (CARA), the demand for insurance will not depend on the fixed cost. In this case, the fixed cost will only affect the decision of whether or not to buy insurance, 14 In our experiment, the subjects did not have the option of over-insuring. 15 In other words, if the risk aversion coefficient is decreasing (DARA), a rise in the fixed cost induces a poverty effect that should increase the marginal indemnity of insurance. So the demand for insurance the average indemnity purchased should increase. 27

29 through a comparison of the utility level observed without insurance. So β 3, β 4 et β 5 should not be significant. Results of the estimation are presented in Table 9, column (2). We observe that, in fact, neither the below-actuarial unit price, nor the cost, nor the fixed-cost cross effects are significant. Thus, insurance is not an inferior good. The only significant effect is when the unit price is above its actuarial value. These results confirm our previous t-tests: since demand is conditional, the fixed cost has no significant impact; nor does the belowactuarial price, regardless of the risk classification of the individuals. 5.3 Partial or full insurance model Finally, in column (3) we examine the determinants of the decision to opt for partial or full insurance conditional on the decision to buy insurance. We run a linear probability model on panel data corrected for selection bias.16 The model specification reflects that of the insurance model, and it yields essentially the same results. The only significant effect results from the above-actuarial price, which reduces the probability of choosing complete coverage. 6 Discussion and Conclusion While the theory of insurance emphasizes the incompatibility between loving risk and buying insurance, our experimental study shows that risk lovers are many and prone to buy insurance under some circumstances. Our experimental approach is a comprehensive laboratory test of the theory of insurance demand. The novelty of our experimental design is twofold: it is based on an original contractual mechanism (a two-part premium) as well as on the taking into account of attitudes toward risk. As regards the contractual mechanism, we resorted to a two-part premium, involving a fixed cost and a unit insurance price, because this contractual arrangement fits well with a reality that is familiar to economic agents. For instance, most health insurance plans consist of a fixed payment (in the form of payroll deductions or taxes) in addition to either variable fees or user fees that reflect the actual level of consumption. Similarly, as a general approximation, we find a two-part structure in most insurance contracts that 16 The estimation of a bivariate probit model with selection bias in the context of panel data raises a plethora of issues. 28