Multidimensional Private Information, Market Structure and Insurance Markets

Transcription

1 Multidiensional Private Inforation, Maret Structure and Insurance Marets Haning Fang Zenan Wu January 17, 2018 Abstract A large epirical literature found that the correlation between insurance purchase and ex post realization of ris is often statistically insignificant or negative. This is inconsistent with the predictions fro the classic odels of insurance a la Aerlof 1970, Pauly 1974 and Rothschild and Stiglitz 1976 where consuers have one-diensional heterogeneity in their ris types. It is suggested that selection based on ultidiensional private inforation, e.g., riss and ris preference types, ay be able to explain the epirical findings. In this paper, we investigate whether selection based on ultidiensional private inforation in riss and ris preferences can, under different aret structures, result in a negative correlation in equilibriu between insurance coverage and ex post realization of ris. We show that if the insurance aret is perfectly copetitive, selection based on ultidiensional private inforation does not result in the negative correlation property in equilibriu, unless there is a sufficiently high loading factor e.g., adinistrative costs. If the insurance aret is onopolistic, however, we show that it is possible to generate the negative correlation property in equilibriu when ris and ris preference types are sufficiently negative dependent, a notion we foralize using the concept of copula. We also clarify the connections between soe of the iportant concepts such as adverse/advantageous selection and positive/negative correlation property. Keywords: Asyetric Inforation; Multidiensional Private Inforation; Adverse Selection; Advantageous Selection; Positive Correlation Property JEL Classification Codes: D82, G22, I11. We are grateful to Mar Arstrong the Editor and two anonyous referees for very detailed coents that significantly iproved the paper. We would lie to than Eduardo Azevedo, David de Meza, Daniel Gottlieb, Ben Lester, Stephen Morris, Yeneng Sun, Veny Venateswaran, Glen Weyl, and seinar/conference participants at National University of Singapore, Monash University, Rice University, NBER Insurance Woring Group Conference 2017 and Asian Meeting of the Econoetric Society 2017 for helpful discussions, suggestions and coents. Part of Fang s research on this project is funded by the generous financial support fro NSF Grant SES Wu thans School of Econoics at Peing University for research support. All reaining errors are our own. Departent of Econoics, University of Pennsylvania, 3718 Locust Wal, Philadelphia, PA 19104; and the NBER. Eail: haning.fang@econ.upenn.edu School of Econoics, Peing University, Beijing, China. Eail: zenan@pu.edu.cn

2 1 Introduction The classic asyetric inforation odels of insurance pioneered by Arrow 1963, Pauly 1974, Rothschild and Stiglitz 1976 and Wilson 1977 assue that potential insurance buyers have one-diensional private inforation regarding their ris type. These odels predict a positive correlation between insurance coverage and ex post realizations of losses. The reason is ex ante adverse selection, naely, that the bad riss i.e., those relatively liely to suffer a loss have a higher willingness to pay for insurance; and allowing for ex post oral hazard only strengthens the positive correlation between coverage and ex post losses. This positive correlation property of the classic asyetric inforation odels fors the basis for epirical tests of asyetric inforation in several recent papers see Chiappori and Salanié However, the results fro a growing epirical literature testing for the correlation between insurance coverage and ex post realization of riss are ixed and vary by aret. In an auto insurance aret, Chiappori and Salanié 2000 find that the accident rate for young French drivers who choose coprehensive autoobile insurance is not statistically different fro those opting for the legal iniu coverage, after controlling for consuers characteristics observable to the autoobile insurers. In contrast, Cohen 2005, using data fro an online Israeli insurer, finds that new auto insurance custoers choosing a low deductible contract tend to have ore accidents, leading to higher total losses for the insurer. 1 In the life insurance aret, Cawley and Philipson 1999 find that the ortality rate of U.S. ales who purchase life insurance is below that of the uninsured, even when controlling for any factors such as incoe that ay be correlated with life expectancy. 2 For the long ter care LTC insurance aret, Finelstein and McGarry 2006, using panel data fro a saple of Aericans born before 1923 the AHEAD study, find no statistically significant correlation between their LTC coverage in 1995 and their use of nursing hoe care between , even after controlling for the insurers assessent of a person s ris type. Moreover, when Finelstein and McGarry 2006 use whether respondents undertae various types of preventive health care as a proxy for ris aversion, they find that people who are ore ris averse by this easure are both ore liely to own LTC insurance and less liely to enter a nursing hoe. In an annuity insurance aret, Finelstein and Poterba 2004 find systeatic relationships between the ex post ortality and the annuity characteristics, such as the tiing of payents and the possibility of payents to the annuitants estate, but they do not find evidence of substantive ortality differences by annuity size. For the Medigap insurance aret, Fang et al find that, conditional on controls for Medigap prices, those with Medigap spend on average $4,000 less on edical care than those without, providing a strong evidence for the negative correlation between Medigap purchase and ex post realization of ris. 1 Others have exained the evidence of asyetric inforation in the choice of insurance contracts such as deductibles and co-payents etc. For exaple, Puelz and Snow 1994 study autoobile collision insurance and argue that, in an adverse selection equilibriu, individuals with lower ris will choose a contract with a higher deductible, and contracts with higher deductibles should be associated with lower average prices for coverage. They find evidence in support of each of these predictions using data fro an autoobile insurer in Georgia. However, see Chiappori and Salanié 2000 and Dionne et al for critiques of the Puelz and Snow study. 2 See He 2009 for a re-exaination of the evidence. 1

3 These epirical findings fueled an eerging literature on the possibility that ultidiensional private inforation ay lead to what has been called advantageous selection. 3 The foral theoretical literature is sparse. de Meza and Webb 2001 postulate a odel in which individuals differ in their ris preferences, which they refer to as tiid and bold types. They assue that ore tiid types ay lower their ris exposure through increased insurance purchase and greater precautionary effort to reduce riss. They show that, in the presence of adinistrative costs in processing clais and issuing policies, there exists a pure-strategy, partial pooling, subgae-perfect Nash equilibriu in the insurance aret that exhibits the negative correlation property. Thus, failure to condition on ris aversion ay then as the positive correlation between insurance coverage and ex post ris predicted by one-diensional odels. Following de Meza and Webb 2001, the existing literature points to ris preferences as the priary suspect behind advantageous selection. In general, however, any private inforation could function as a source of advantageous selection if it is positively correlated with insurance coverage and at the sae tie negatively correlated with ris. Finelstein and McGarry 2006 argue that their findings on the LTC insurance aret is consistent with ultidiensional private inforation and advantageous selection based on ris aversion. In fact, their findings suggest that, on net, adverse selection based on ris and advantageous selection based on ris aversion roughly cancel out in the LTC insurance aret. Fang et al find that, for Medigap insurance aret, ris preferences do not appear as a source of advantageous selection, but cognitive ability is particularly iportant. However, to the best of our nowledge, the precise conditions under which whether selection based on ultidiensional private inforation ay generate in equilibriu a positive or negative correlation between insurance purchase and ex post realization of ris is still unnown. Most of the existing papers that invoed the possibility of ultidiensional private inforation as a possible explanation for the epirical findings discussed above rely on partial equilibriu intuition uch in the spirit of Heenway, An iportant exception is Chiappori, Jullien, Salanié and Salanié 2006, henceforth CJSS, which argues that in a copetitive insurance aret the positive correlation property is a general iplication of insurance odels with asyetric inforation even when the private inforation is ultidiensional in riss and ris preferences. The ey assuptions are consuer rationality and a condition which they refer to as nonincreasing profit NIP condition - that is, the per contract expected profit does not increase with the coverage of the contract. 4 CJSS s approach is general and elegant, and they prove their results using revealed preference and the NIP condition. However, the non-increasing profit condition is not a priitive condition; thus, whether it holds in equilibriu in environents where the aret ay not be copetitive and where loading costs for offering insurance exist is still an open question. The goal of this paper is to help fill in this gap. We present a siple odel of insurance aret where consuers have ultidiensional private inforation in ris and ris preference types, and investigate whether selection based on ultidiensional private inforation can, under different 3 The first description of this phenoenon in the econoics literature appears to be Heenway 1990, who used the ter propitious selection. 4 We will discuss the connection between our results and theirs in Section 4. 2

4 aret structures, result in negative correlation in equilibriu between insurance coverage and ex post realization of ris. We show that if the insurance aret is perfectly copetitive, selection based on ultidiensional private inforation does not generate the negative correlation property in equilibriu unless there is a sufficiently high loading factor, possibly because of adinistrative or areting costs. If the insurance aret is onopolistic, however, we show that it is possible to generate the negative correlation property in equilibriu when consuers ris type and ris preference type are sufficiently negative dependent, a notion we foralize using the concept of copula. It should be noted that the fully specified odel considered in our paper is not as general as that in CJSS; our contribution is to state the connections between the priitives of the odel including ultidensional private inforation and aret structure and the positive or negative correlation property in a transparent way. The reainder of the paper is structured as follows. In Section 2 we provide a detailed discussion of the related literature. In Section 3, we describe our odel environent in which consuers are heterogeneous in both ris and ris preference types. In Section 4, we consider the perfectly copetitive aret structure. In Section 5, we analyze the onopolistic aret. 5 In Section 6, we clarify the confusions in this growing literature about the connections between soe of the iportant concepts such as adverse/advantageous selection and positive/negative correlation property. In Section 7, we partially endogenize the contract space and again show that our results for the single contract case derived in Sections 4 and 5 continue to hold with natural and ild generalizations of the assuptions iposed in Section 3. In Section 8, we suarize our ain findings and suggest directions for future research. All proofs are relegated to an Appendix. 2 Related Literature To the extent that our paper investigates on whether the positive correlation property is robust to environents with ultidiensional consuer heterogeneity, it is ost related to CJSS 2006 and de Meza and Webb CJSS argue that, as long as consuers are rational and the per contract expected profit does not increase with the coverage of the contract which they refer to as nonincreasing profit NIP condition, then the positive correlation property is robust to ultidiensional private inforation. This conclusion is siilar to our results for the copetitive insurance aret presented in Propositions 1-2 and Proposition 7. Their results are proved using the revealed preference arguent iplied by the hypothesized consuer rationality and the nonincreasing profit condition assued on the supply side, and as such they do not have to exploit the full set of equilibriu restrictions. In contrast, we exploit the full set of the equilibriu restrictions and as a result our positive correlation predictions for the copetitive insurance aret are sharper for the case of proportional loading factor. We will provide ore details of the coparison when we discuss Proposition 2. Also related, de Meza and Webb 2017 provide an insightful discussion that the positive correlation test is not a valid test to distinguish asyetric inforation fro 5 In an online appendix, we show that our results for the onopolistic aret structure can be generalized to a version of an iperfectly copetitive aret structure. 3

5 syetric inforation environent. The reason is that under syetric inforation, the only insurance purchased by consuers will be the one with full coverage, unless the clai processing costs or other loading factors are forally odeled. The results in our paper are ore relevant in distinguishing ulti- vs. one-diensional private inforation odels of insurance as opposed to syetric vs. asyetric inforation odels of insurance. Our paper is also related to a recent literature that attepts to analyze the selection arets with potentially ultidiensional private inforation. Einav et al. 2010b propose an approach to conduct epirical welfare analysis in insurance arets based on directly estiating the deand and average and arginal cost curves using exogenous variations in prices. 6 Based on their graphical analysis of the deand and cost curves for the selection aret, where the defining feature is that insurers costs depend on which consuers purchase their products and hence are endogenous to price, they also argue that the slope of the estiated arginal cost curve provides a direct test of the existence and the nature of selection, i.e., whether the selection is adverse or advantageous. Specifically, they argue that a rejection of the null hypothesis of a constant arginal cost curve is a rejection of the null hypothesis of no selection, whereas the selection is adverse respectively, advantageous if the arginal cost is increasing respectively, decreasing in price. They also ephasize that an attractive feature of their approach of relying only on the estiated deand and cost curves is that it does not require the researcher to ae often difficult-to-test assuptions about consuers preferences or the nature of ex ante inforation Einav et al., 2010b, p In Section 6, we show that an iportant liitation of their approach is that typically the arginal cost curve is non-onotonic when consuer heterogeneity is ultidiensional. 7 In fact, in our setting with two-diensional private inforation in ris and ris aversion, it is onotonic only when the two diensions are perfectly correlated. Therefore, depending on the range of the price variations available in the data that is used to estiate the deand and cost curves, it is liely the estiated cost curve only reflects the nature of the selection adverse or advantageous locally. 8 We also provide an exaple Exaple 4 in which the nature of the local selection being advantageous at the equilibriu price level does not iply a negative correlation between insurance purchase and ex post realization of ris in equilibriu. We explicitly odel consuers ultidiensional heterogeneity in this paper. The intuition for why the arginal cost curve is unliely to be onotonically increasing in the aret size in our odel can be easily explained for the case of bounded consuer type space. Consider an environent where consuers willingness to pay for insurance is increasing in their ris type and ris ] aversion type λ. Suppose that ris type and ris aversion type are bounded in, ] and λ, λ respectively. Then the arginal cost of insurance when the aret size is close to 0 will 6 See also Einav et al. 2010a, Einav and Finelstein 2011 and Chetty and Finelstein 2013 for related discussions of the deand and cost analysis of selection arets. 7 Einav and Finelstein 2011 are aware of the non-onotonicity issue of the arginal cost curve, as they stated: More generally, once we allow for preference heterogeneity, the arginal cost curve needs not be onotone. However, for siplicity and clarity we focus our discussion on the polar cases of onotone cost curves. footnote 7, p In epirical applications, the range of the exogenous price variations is often quite liited. For exaple, Einav et al. 2010b have a total of six price levels or three price levels if one only considers those with soewhat large nuber of consuers. 4

6 be close to and the arginal cost when the aret size is close to 1 will be close to. That is, the arginal cost curve ust always have at least one decreasing segent! Another benefit of odeling consuers ultidiensional heterogeneity explicitly is that it allows us to exaine how the aret outcoe changes when the dependence structure of individuals ultidiensional heterogeneity varies. We use the concept of copula to paraeterize the degree of dependence between the consuers ultidiensional types. Mahoney and Weyl 2017 build and analyze a odel of iperfect copetition in selection arets. They paraeterize the degree of both aret power and selection, and use graphical price-theoretic reasoning to analyze the interactions between selection and iperfect copetition. Their paraeterization of selection follows Einav et al. 2010b by hypothesizing whether the arginal cost curve is either upward or downward sloping. Azevedo and Gottlieb 2017 propose an equilibriu concept for copetitive insurance aret where consuers ay have ultidiensional heterogeneity, and insurance copanies copete for consuers by choosing contracts fro a copact space and setting their corresponding prices. Their equilibriu concept, which relies on perturbations, guarantees existence. Veiga and Weyl 2016 instead study the incentives for a onopolistic insurer in its choice of insurance quality facing consuers with ultidiensional heterogeneity. They derive a condition of the optial insurance quality to ephasize that the sorting incentives of the onopolist is the ratio of two ters: the nuerator is the covariance aong arginal consuers between the arginal willingness to pay for quality and the cost for the fir, and the denoinator is arginal consuer surplus, which easures aret power. The analysis of Veiga and Weyl 2016 focuses on the arginal consuers, and does not analyze correlation between insurance purchase and ex post realization of ris which is about the average insurance buyers and non-buyers; and they also focus on the onopolistic aret structure, while our paper highlights the interactions between ultidiensional heterogeneity and aret structure. 9 In the basic odel of our paper, the quality of insurance is assued to be exogenously fixed, and we focus on the deterination of preiu; while Veiga and Weyl 2016 focus on how the onopolistic insurer deterines the profit-axiizing quality of insurance. Neither Azevedo and Gottlieb 2017 nor Veiga and Weyl 2016 analyze whether ultidiensional consuer heterogeneity can generate a negative correlation between insurance purchase and ex post realization of ris in equilibriu The Model Consuers. There is a continuu of consuers with heterogeneous types indexed by θ, λ, where, ] with 0 < < denotes consuer s ris type, and λ λ, λ] with 9 Also related to our paper, Weyl and Veiga 2014 offer a quantitative strengthening of the notion of affiliation for ultidiensional rando vectors that is useful to relate dependence between ris types and ris preferences to the direction of selection. 10 It should be noted that Azevedo and Gottlieb 2017 introduce a notion of intensive argin selection coefficient that easures the difference between the arginal changes of the preiu and the cost of insuring the arginal consuers, both with respect to the insurance coverage. They suggest that this notion is related to the positive correlation test. We will discuss its connection with our results in Sections 6 and 7 below. 5

7 0 < λ < λ < can be interpreted as any other characteristics of the consuer that ay be related to his/her ris preference. 11 As a notational convention, we use M and Λ respectively to denote the rando variables for ris type and ris preference type, and their lowercase counterparts as their realizations. In the population, consuers type,, λ, is assued to be drawn fro joint CDF H, λ, and we denote the arginal CDF of M and Λ by F and G λ respectively. We use h, λ, f and g λ to denote the corresponding joint and arginal density functions of M, Λ, M and Λ, respectively. We assue that the arginal distribution of M is such that M has finite ean, denoted by E M]. We allow dependence between M and Λ in this paper, and we will discuss the for of the dependence in detail in Section 5. Insurance Contract. Consuers decide whether or not to purchase insurance. In the basic odel, we assue that insurance firs are regulated in the sense that they can only provide insurance with quality x 0, 1] where a higher x indicates a contract with better coverage. Note that we assue that the insurance coverage quality x is not a choice variable for the firs. As such, our setup is in the spirit of Aerlof 1970 where insurance contract is exogenously given, rather than Rothschild and Stiglitz 1976 where the insurance quality x is endogenously chosen. While exogenous contract space is an iportant restriction, it is useful to point out that this approach is adopted in ost of the recent applied literature. For instance, Einav et al. 2010b assue that a consuer either buys a hoogeneous insurance policy or does not buy any coverage. Siilarly, Handel et al assue that there are only two options: a low-coverage contract and a high-coverage contract. The cost to the insurance firs, not including loading costs e.g., contract processing cost and other adinistrative costs, for providing quality-x insurance to a type-θ, λ consuer, denoted by C θ; x, is increasing in the consuer s ris type and the coverage quality x. Note that it does not depend on the consuer s ris preference type λ. In particular, we let Cθ; x = x. Consuer Preference. Type-θ, λ consuer derives utility Uθ; x, p fro purchasing a contract with quality x 0, 1] and preiu p 0,. Without loss of generality, we assue that consuers derive zero utility fro the null contract x, p = 0, 0, that is, Uθ; 0, 0 0 for all θ. Assuption 1 U/ p < 0. Moreover, fixing x 0, 1], there exist p and p such that Uθ; x, p < 0 < Uθ; x, p. Denote type-θ, λ consuer s willingness to pay WTP for an insurance policy with quality x by vθ; x. By definition, vθ; x is the solution to Uθ; x, v = Uθ; 0, 0 = We assue that and λ are bounded above by and λ respectively for the siplicity in describing soe of the intuitions for our results. Most of our results reain qualitatively unchanged if ris type and ris preference type are supported on sei-infinite intervals, that is,, λ, λ,. In particular, see our discussion on the differences caused by the sei-infinite support in Footnote 32. 6

8 Assuption 1 guarantees the existence and uniqueness of such a solution for all x 0, 1]. We ae the following assuptions on v : Assuption 2 v/ x > 0, v/ > 0, and v/ λ > 0. Assuption 2 siply says that consuer s WTP for insurance is increasing in her ris type, in her ris preference type and the quality of the contract coverage. Assuption 3 vθ; x > Cθ; x x, for all θ and any x 0, 1]. Assuption 3 holds for any econoic fraewor of insurance as long as individuals are risaverse. 12 The difference between the WTP for insurance vθ; x and Cθ; x is coonly referred to as the ris preiu for type-, λ consuer. Facing a preiu p for insurance coverage x, a type-, λ consuer purchases insurance if and only if Uθ; x, p Uθ; 0, 0 0, or equivalently, p v θ; x. We use B p { θ : vθ; x p } 2 to denote the set of consuers whose WTP for the insurance exceeds the preiu and thus they are the set of buyers; and use to denote the set of non-buyers at price p. N B p { θ : vθ; x < p } 3 Assuption 2 iplies that the iso-wtp curve is downward sloping in the, λ space for x 0, 1] and the set of consuers above respectively, below the curve is the set of buyers respectively, non-buyers. In addition, Assuption 1 together with Uθ; 0, 0 = 0 iplies that consuers WTP for zero coverage is zero, that is, vθ; 0 = 0 for all θ. Rear 1 In practice, fir can charge preius based on observable characteristics. In this paper we siplify our analysis by assuing the observed characteristics are the sae across all consuers. It is useful to thin of our analysis as being within the consuers of a particular ris classification class. This siplification allows us to focus on the coparison between ultidiensional and onediensional private inforation. Rear 2 We would lie to ephasize that the ey distinction between one-diensional and ultidiensional private inforation odels is whether the raning of consuers by their WTP is perfectly aligned with the raning by their costs. Assuption 2 iplies that the two ranings are not necessarily perfectly aligned in our odel, whereas they are always perfectly aligned in the classic asyetric inforation odels of insurance with one-diensional private inforation in ris types. It is in this sense that our odel is ultidiensional. It should also be noted that, at a 12 In an environent with heterogeneity in ex post oral hazard, such as that studied in Einav et al. 2013, it is possible that v θ; x does not always exceed C θ; x, as shown in Exaple 3 of Azevedo and Gottlieb Moral hazard ay also indirectly lead to a violation of Assuption 2 if consuers face budget constraints. 7

9 technical level, it is always possible to encode ultidiensional types in a single-diensional variable, so, as always, the content of the ultidiensional signals depends on additional assuptions ade about the type space. 13 Exaple 1 Binary States Each consuer has initial wealth y and is subject to a possible loss ω 0, y with probability κ. The consuer can purchase an insurance contract to cover a fraction x of the loss if it occurs. Let u ; λ be consuer s Bernoulli utility function where λ is the ris preference paraeter. Let κω. Then the expected cost to the insurance fir for insuring type-, λ consuer is C, λ; x = xκω = x. Consuers net expected utility fro purchasing an insurance x, p is Uθ; x, p ω u y p 1 xω; λ + 1 ] uy p; λ ω uy ω; λ + ω 1 ] uy; λ. ω It is straightforward to verify that Assuption 1 is satisfied and Uθ, 0, 0 = 0 for all θ. Consuers WTP for insurance of coverage x is deterined by Uθ; x, v = 0, or equivalently, ω u y v 1 xω; λ + 1 uy v; λ = ω ω uy ω; λ + 1 uy; λ. ω By the iplicit function theore, it can be verified that v/ x > 0 and v/ > 0. In addition, v/ λ > 0 holds if λ orders types according to their ris aversion in the sense of Pratt 1964: λ 1 > λ 0 u x, λ 1 u x, λ 1 u x, λ 0 u x, λ 0 x. Therefore, Assuption 2 is satisfied. Lastly, the concavity of u ; λ iplies that vθ; x > x = C θ; x and hence Assuption 3 is also satisfied. Exaple 2 CARA and Noral Shocs Consuers have initial wealth y and ay experience a edical expenditure Z N, σ 2. A consuer has constant absolute ris aversion CARA Bernoulli utility uy = exp λy, where λ is consuer s constant absolute ris aversion. With CARA Bernoulli utility, it is without loss of generality to easure consuer s utility by his certainty equivalent, that is, Solving Equation 1 for v yields, vθ; x = x + Uθ; x, p x + x 2 x σ 2 λ p. 2 x 2 x σ 2 λ = C θ; x + 2 x 2 x σ 2 λ Multidiensional types can always be encoded in a single-diensional variable using the inverse Peano function e.g., Sagan 1984, p. 36] and other ethods. The difficulty of such a one-diensional representation of an intrinsically ultidiensional proble, however, is that we could not ipose reasonable restrictions on the inforation structure, such as types being drawn fro a continuous distribution. Siilar issues concerning the representation of ultidiensional inforation with single-diensional essages have been discussed in the echanis design literature see, e.g., Mount and Reiter Also see Fang and Morris 2006 for siilar discussions. 8

10 It can be verified that Assuptions 1-3 are satisfied in Exaples 1 and Copetitive Insurance Maret In a copetitive insurance aret, insurance firs choose a preiu p for insurance coverage with the given quality x to copete for consuers. A fir s profit at preiu p fro offering insurance with coverage x, if there is no loading cost of offering insurance, is given by: 15 πp = θ Bp p x dh, λ. 5 Denote p as the equilibriu price under perfect copetition which, in the absence of loading costs, is siply deterined by: πp = 0. 6 Rear 3 Equation 6 ay have ultiple solutions. Any preiu greater than v, λ; x satisfies 6 because B p is epty for any p > v, λ; x. If consuers ris preiu is sufficiently large, it is also possible that the equilibriu preiu is less than v, λ; x ; in such a scenario, all consuers purchase insurance. Because our goal is to copare the average ris of the consuers with insurance and those without, we assue in the rest of the paper that there exists at least one equilibriu with preiu p that lies strictly between v, λ; x and v, λ; x so that the sets B p and N B p are of positive easures to ensure the conditional expectations 7 and 8 below are both well defined. Fixing the aret price of the insurance p v, λ; x, v, λ; x, the average ex post realization of ris aong those who purchase insurance is: E M B p ] = θ Bp θ Bp dh, λ, 7 dh, λ where the denoinator is the easure of the insurance coverage penetration, and the nuerator is the total cost realization of the insured. Siilarly, the average ex post realization of ris aong those who do not purchase insurance is E M N B p ] = θ N Bp θ N Bp dh, λ. 8 dh, λ 14 See Online Appendix A for the details of the proof that Exaples 1 and 2 satisfy Assuptions We will consider how loading costs affect our results below. 9

11 It is also useful to define the average ex post realization of the ris for the entire population: E M] = = λ λ θ Bp dh, λ dh, λe B p ] + dh, λe N B p ]. 9 θ N Bp Definition 1 Positive and Negative Correlation Property The insurance aret exhibits positive correlation property in equilibriu if E M B p ] > E M N B p ], and it exhibits negative correlation property if E M B p ] < E M N B p ], where the two ters are defined in 7 and 8 respectively. Note that Definition 1 defines the positive and negative correlation property for the case of positive coverage versus zero coverage. It is straightforward to generalize our results below to the case where the coparisons are between the expected cost realizations of high versus low coverage. In Section 7 we will forally generalize the definition of positive and negative correlation property when there are contracts with ultiple levels of coverages. Proposition 1 Positive Correlation Property Always Holds in Copetitive Equilibriu without Loadings Suppose Assuptions 1, 2 and 3 are satisfied and that the equilibriu price p is such that the easure of buyers and non-buyers are both strictly positive. Then positive correlation property always holds in equilibriu if the insurance aret is perfectly copetitive and there are no loadings. Note that Proposition 1 states that negative correlation property will not eerge in a copetitive insurance aret without loadings, regardless of the dependence structure between ris type and ris preference type. The intuition for the result is in fact very siple. If there were a negative correlation between insurance purchase and ex post ris realizations, then in a copetitive insurance aret, the equilibriu preiu ust be equal to the expected ris realization of the insured, which is lower than that of the uninsured under negative correlation property. But if the equilibriu preiu were indeed lower than the expected ris realization of the uninsured, it ust also be lower than their average WTP under Assuption 3. This in turn iplies that at the equilibriu preiu, soe of the uninsured ust prefer to purchase insurance as well, which is a contradiction. The following siple three-type exaple illustrates the aforeentioned intuition. Exaple 3 Illustrative Three-Type Exaple Suppose that there are three types of consuers in the population. For siplicity, we will describe their types by the cobinations of their cost of coverage and WTP for insurance: c 1, v 1, c 2, v 2 and c 3, v 3. Assuption 3 iplies that v j > c j for j {1, 2, 3}. Let q j denote the probability that a consuer is of type c j, v j, j {1, 2, 3}, in the population. Suppose that c 1 < c 2 < c 3. In the standard one-diensional private inforation odel, consuers differ only regarding their ris types. Hence the discrete analog of Assuption 2 would iply that v 1 < v 2 < v 3. Thus, it is iediate that the positive correlation property ust hold in any copetitive equilibriu. 10

12 In a ultidiensional private inforation odel, the order of consuers WTP ay differ fro the order of their ris types or their costs of coverages due to the possibility that consuers with lower ris ay be ore ris averse. In order for a pure-strategy copetitive equilibriu to exhibit the negative correlation property, there are only three possibilities: 16 i only type c 1, v 1 consuers buy coverage in equilibriu at a preiu p = c 1 ; ii type c 1, v 1 and type c 2, v 2 consuers purchase coverage in equilibriu at a preiu p = q 1 q 1 +q 2 c 1 + q 2 q 1 +q 2 c 2 and type c 3, v 3 consuers reain uninsured; iii type c 1, v 1 and type c 3, v 3 consuers purchase coverage in equilibriu at a preiu p = q 1 q 1 +q 3 c 1 + q 1 q 1 +q 3 c 3 and type c 2, v 2 consuers reain uninsured. We show that none of the cases are possible. For case i, equilibriu requires type c 1, v 1 consuers to prefer to buy coverage and type c 3, v 3 consuers to prefer to reain uninsured, that is, v 1 p = c 1 v 3. The above inequality, together with v 3 > c 3, iplies iediately that c 1 > c 3, a contradiction. Siilarly, for case ii to constitute an equilibriu, we ust have that type c 2, v 2 consuers purchase the policy and type c 3, v 3 consuers choose to opt out, that is, v 2 p = q 1 q 1 + q 2 c 1 + q 2 q 1 + q 2 c 2 v 3. The above inequality, together with v 3 > c 3, iplies iediately that cannot be satisfied due to the postulated c 1 < c 2 < c 3. q 1 q 1 +q 2 c 1 + q 2 q 1 +q 2 c 2 > c 3, which For case iii, in order for the equilibriu to have negative correlation property, we need to have p = q 1 q 1 + q 3 c 1 + q 1 q 1 + q 3 c 3 < c 2, but then it iplies v 2 > p because v 2 > c 2, i.e., type c 2, v 2 consuer will also buy insurance, which is a contradiction. Now we consider the role of loading factors, which include both underwriting-based loading and clai-based loading. Denote the loading factor by l > 0, and by p l the copetitive equilibriu price in a aret with the loading factor l, which is deterined by: p l = 1 + l xe M B p l ]. 10 Define EM 11 as the average ris type in the population of consuers. We have the following result: It is straightforward to generalize the arguent to allow for ixed strategies. 17 If the loading factor is additive instead of ultiplicative, i.e., if the equilibriu preiu satisfies p l = xe M B p l ] + l, 11

13 Proposition 2 Positive Correlation Property Holds in Copetitive Equilibriu with Low Loadings Suppose Assuptions 1, 2 and 3 are satisfied. A sufficient condition for the positive correlation property to hold in equilibriu if the insurance aret is perfectly copetitive is l v, λ; x x Note that the upper bound on the loading factor specified by 12 depends on the ratio of the WTP relative to its expected clai for a consuer who has the average ris and the lower-bound ris preference. Proposition 2 shows that if the loading factor is bounded above by 12, then the copetitive insurance aret will always exhibit a positive correlation property in equilibriu even in the presence of loading factors. The intuition is again quite siple. In order for an equilibriu that the low ris types purchase insurance, while the high ris types do not, to exist, it ust be the case that the preiu is higher than the WTP for the high ris types. But the only way for such levels of preiu to be consistent with equilibriu when the insured is actually of low ris types is that the loading factor is very high, which is ruled out by the upper bound 12 on the loading factor. Rear 4 We could have redefined consuers ris type to be inclusive of the insurance loadings. Such a redefinition of ris type will ae Assuption 3 ore stringent. The sufficient condition 12 stated in Proposition 2 requires the ris preiu for the average ris type to be sufficiently high. To illustrate why the stated upper bound on loading factors is sufficient to rule out the negative correlation property in a copetitive equilibriu, we introduce loadings into Exaple 3. Exaple 3 Continued, Ipossibility of Negative Correlation Property with Low Loadings Suppose that c 2 = j {1,2,3} q jc j = E M]; in words, c 2 is the average ris type in the population of consuers, as defined in 11. Now suppose that the aret equilibriu price p is such that only type c 1, v 1 consuers are purchasing insurance, i.e., that the aret equilibriu exhibits negative correlation property. For this to be an equilibriu, it ust be the case that the equilibriu price p = 1 + l c 1 and it satisfies ax {v 2, v 3 } < p v However, this inequality is ruled out by the upper bound on the loadings in 12, which for this exaple is reduced to then the corresponding sufficient condition for Proposition 2 is l v, λ; x x. 1 + l v 2 c

14 To see this, note that 14 iplies that p = 1 + l c 1 v 2 c 2 c 1 < v 2, 15 which is a contradiction against 13, iplying that type c 2, v 2 consuers would have preferred to purchase insurance as well at p. Proposition 2 states that the positive correlation property is robust to a sufficiently sall loading factor. However, when the loading factor is large enough, it is possible to obtain the negative correlation property. Again we continue with Exaple 3 to elaborate this point. Exaple 3 Continued, Possibility of Negative Correlation Property with Moderate Loadings Suppose for siplicity that q 2 = 0 and thus there are two types of consuers. In order for the aret to exhibit the negative correlation property, we ust have that type c 1, v 1 consuers purchase the insurance policy whereas type c 3, v 3 consuers reain uninsured, that is, which is possible if and only if v 3 p = 1 + lc 1 v 1, v 3 c 1 1 l v 1 c Therefore, if the load is higher than v 3 c 1 1 but lower than v 1 c 1 1, there exists an equilibriu in which negative correlation property holds. Note that 16 indicates that a necessary condition for negative correlation property to eerge is that v 1 v 3, which cannot be satisfied with one-diensional consuer heterogeneity in ris types. The logic underlying Propositions 1 and 2 suggests that whether ultidiensional private inforation, particularly private inforation related to ris preferences, can explain the observed negative correlation between insurance purchase and ex post ris realization ust be related to large loading factors, or soe non-copetitive features of the insurance aret. 18 The iportance of the size of the loading factor highlighted in Propositions 1 and 2 suggests that the different findings in Chiappori and Salanié 2000 and Cohen 2005 we entioned in the introduction can potentially result fro the differential loading factors in the two arets. Recall that Cohen s 2005 data is fro an online Israeli insurer, while Chiappori and Salanié 2000 is fro a traditional French insurance copany. The online insurer is liely to have a uch lower loading than the traditional insurer. Therefore Proposition 2 suggests that positive correlation property is ore liely to hold in the Israeli data. Systeatic epirical studies of the size of the loading factor are rare. de Meza and Webb 2001, p. 250 note that Between 1985 and 1995 for U.K. insurers, expenses as a percentage of preiu incoe averaged 25% for otor insurance and 18 This constrasts with a view that was shared by any in the literature. For exaple, Chetty and Finelstein 2013, p.125 stated that... if preferences are sufficiently iportant deterinants of deand for insurance and sufficiently negatively correlated with ris type, the aret can exhibit what has coe to be called advantageous selection. See, however, CJSS 2006, p

15 37% for property daage insurance. The Affordable Care Act in the United States regulates that health insurers need to aintain a iniu edical loss ratio the fraction of preiu that need to be used to pay for clais of 80 percent, which iplies a loading factor of no ore than 25% since 80% = 1/ % ]. 19 It is useful to point out that such levels of average expense/preiu ratio can still be consistent with the sufficient condition stipulated in Proposition 2, which is a condition iposed only on the average ris type as defined by 12. Relationship to CJSS 2006 Let us now discuss in details the connections between our Propositions 1-2 and the results in CJSS Forally, using the notation in CJSS, a contract C i reiburses the insured an aount R i L when loss L occurs, and they say that contract C 2 covers ore than contract C 1 if R 2 L R 1 L is nondecreasing in L. Let π C i = p i R i L df i L Γ denote the per contract profit fro contract C i where F i L is the distribution of loss aong consuers purchasing contract C i, and Γ is the fixed costs associated with the contract, and it is assued to be the sae across the contracts. The nonincreasing profit NIP condition states that π C 2 π C 1 if contract C 2 covers ore than C 1. CJSS s ain result, Proposition 2 p. 789, states that under assuptions on consuer rationality which we also assue and the nonincreasing profit condition, it ust be true that R 2 L df 2 L R 2 L df 1 L. 17 In the notation of our paper, the case we considered in Proposition 1 is the case with Γ = 0, R 2 L = xl and R 1 L = 0 where L = M. 20 The NIP condition is autoatically satisfied in our copetitive aret setting because of the zero profit condition. Under this interpretation, CJSS s inequality 17 is equivalent to E M B p ] E M N B p ]. Thus, our result as stated in Proposition 1 is consistent with CJSS s inequality 17. then When there is a proportional loading factor l > 0, the per contract profit fro contract C i is π C i = p i 1 + l R i L df i L, and CJSS show that their testable iplication is given by their inequality 7 on p. 790, with tax 19 See 20 When Γ > 0, CJSS s Proposition 1 can not be used to show that the average ex post realization of ris for the insured is higher than the uninsured, as CJSS pointed out in their Footnote 5 p

16 rate t = 0: 21 R 2 L df 2 L R 2 L df 1 L l R 1 L df 1 L ] R 2 L df 2 L. 18 Again, if we let R 2 L = xl and R 1 L = 0 where L = M to atch the setting considered in our Proposition 2, the inequality 18 can be siplified as E M B p ] E M N B p ] le M B p ], which does not correspond to the positive correlation property even if the loading factor l is sufficiently sall. Indeed, CJSS coent that p. 791,... we can test soe well-defined iplication of asyetric inforation which ay not loo lie a positive correlation property any ore italics added]. We thus believe that our Proposition 2 is copleentary to CJSS 2006 and provides soe new insights to the existing literature. 5 Monopolistic Insurance Maret In this section, we focus on the other extree of the insurance aret structure, assuing that there is a onopolistic insurance fir that chooses a preiu to axiize its profit. 22 We as whether correlated ultidiensional private inforation can lead to the eergence of negative correlation property see Definition 1 in a onopolistic insurance aret. 23 We first provide soe bacground on how we will odel dependence of the two diensional private inforation M and Λ. In Section 3, we stated that, in the population, consuers type,, λ, is independently drawn fro joint CDF H, λ, with arginal CDFs for M and Λ respectively denoted by F and G. It turns out to be easier to paraeterize the dependence structure of the two rando variables M and Λ using the concept of copula. 24 By Slar s Theore, for every joint distribution H, λ, there exists a unique copula C, such that H, λ = CF, Gλ. That is, the dependence structure between M and Λ can be represented by a copula and reains unchanged under strictly increasing transforations of the rando variables. We first consider the case of positive dependence between the ris type M and the ris preference type Λ. Although intuition suggests that the positive correlation between ris and ris preference would exacerbate adverse selection and thus strengthen the positive correlation between insurance coverage and ex post realization of ris, here we provide a precise sufficient condition for such a conclusion. 21 CJSS used the notation E i L] to denote the expected clais under contract C i, and they wrote E i L] = LdFi L. We believe that it is a typo and should be E i L] = R i L df i L. 22 We will allow the onopolistic insurance fir to choose both the preiu and the coverage in Section In Online Appendix B, we introduce a paraeterization of the iperfectly copetitive aret structure and show that our results for the onopoly case are robust. 24 See Nelsen 2006 for an excellent introduction to copulas. 15

17 Definition 2 Positive Stochastic Monotonicity Dependence Λ is stochastically increasing in M if PrΛ > λ M = is a nondecreasing function of for all λ. Nelsen 2006, Corollary , p. 160 proved that Definition 2 is equivalent to C 11 z 1, z 2 0 for all z 1, z 2 0, 1] 2 in the language of copula. Positive stochastic dependence eans that a high realization of z 1 shifts the conditional distribution of z 2 according to first-order stochastic doinance. Because arginal distribution functions are onotonic, this property of copula translates directly into corresponding dependence property of the joint distribution of M, Λ. Proposition 3 Positive Stochastic Monotonicity Dependence Iplies Positive Correlation Property Suppose Assuptions 1, 2 and 3 are satisfied. If Λ is stochastically increasing in M, then positive correlation property holds under onopoly. In fact, the positive correlation property result in Proposition 3 applies to any aret structure. Moreover, it is useful to point out that Proposition 3 is general in the sense that it does not rely on the functional for of consuer s WTP. As long as consuers WTP is increasing in both and λ, Proposition 3 holds. In the rest of the section we focus on the case in which the ris type M and the ris preference type Λ exhibit negative dependence to be ade precise below and investigate whether the negative dependence between M and Λ ay lead to the eergence of negative correlation property under a onopolistic aret structure. We consider the tractable case of CARA utility function and norally distributed shocs as described in Exaple 2. Note that for this CARA-Noral specification, it is without loss of generality to assue that x = 1; 25 thus fro 4, we have Cθ =, and v, λ = + λ where σ We will interpret the paraeter σ 2 /2, where σ 2 is the variance of the health expenditure shoc, as the relative iportance of ris aversion as a deterinant of the consuer s WTP for insurance: a higher σ 2 eans that consuers are subject to ore volatility in health expenditure, and as a result ris aversion becoes ore iportant in deterining the WTP for insurance. The Role of Preferences We study the effect of the relative iportance between riss and preferences, i.e., the agnitude of defined in 19, holding fixed the joint distribution H,. Proposition 4 Suppose that consuers have CARA utility functions and experience norally distributed riss as described in Exaple 2. For every H,, there exists a threshold > 0 such that for all <, E M B p ] > E M] for all p + λ, + λ. Proposition 4 shows that if ris preference is not a sufficiently iportant deterinant of the deand for insurance, then negative correlation property will not eerge under onopolistic aret structure, regardless of the joint distribution, H,, of riss and ris preferences. Notice that this 25 For x 0, 1, we can redefine ˆv, λ ; 1 v, λ ; x/x and Ĉ θ; 1 C θ; x /x. 16

18 holds true even when the ris type M and the ris preference type Λ exhibit strong negative dependence. Figure 1 illustrates why this is so for the extree case when M and Λ exhibit perfect negative dependence. Since M and Λ are perfectly negative dependent, there exists a one-to-one onotonic apping between and λ, which is shown in the dashed line in Figure 1. A sufficiently sall yields a steep iso-wtp curve, which would iply that for any price in + λ, + λ, the iso-wtp curve that separates purchasers and non-purchasers of insurance at that price would intersect the dashed line as depicted: the higher riss the darer segent of the dashed line always purchase insurance before the lower riss the lighter segent of the dashed line. This results in the positive correlation property. λ uninsured λ insured Figure 1: Selection Based on Ris Type: The Case of Low Intuitively, when is sufficiently sall, a change in the price charged by the onopolist will have a stronger influence on the support of ris type M rather than that of Λ; as a result, the aret is ore susceptible to the ris-based adverse selection proble as in the case of one-diensional private inforation in ris, and the potential countervailing effect of selection based on ris preferences is too wea to override the positive correlation property. Now we consider the other liiting case, and show that if ris preference is a sufficiently iportant deterinant for the deand of insurance, then when ris M and ris preference Λ are negatively dependent to be defined ore precisely below, a onopolistic aret ay exhibit negative correlation property in equilibriu. To this end, we first introduce the notion of negative quadrant dependence: 17

19 Definition 3 Strict Negative Quadrant Dependence M and Λ are strictly negatively quadrant dependent if for all, λ, ] λ, λ], H, λ < F Gλ. Strict negative quadrant dependence foralizes the notion that two rando variables are negatively dependent if greater values of M are ore liely to appear with saller values of Λ and vice versa. In the language of copula, it is equivalent to Cz 1, z 2 < z 1 z 2. Notice that independent rando variables does not satisfy strict negative quadrant dependence see Nelsen Proposition 5 Suppose that consuers have CARA utility functions and experience norally distributed riss as described in Exaple 2. If M and Λ are strictly negatively quadrant dependent, then there exists a threshold such that the negative correlation property eerges under onopoly when >. The intuition for Proposition 5 is as follows. When ris aversion is a sufficiently iportant deterinant of the deand for insurance, the iso-wtp curve is sufficiently flat in the, λ space. Thus, a change in the price by the onopolistic fir will have a stronger ipact on the distribution of the ris aversion type than the distribution of ris types aong the set of the purchasers; that is, the selection of the consuers are ore based on ris aversion type λ than on ris type. In the liit when is sufficiently large, it is profit axiizing for the onopolist to price in a way to select only consuers whose ris aversion is above λ arg ax λ1 Gλ]. Because higher ris aversion is associated with lower ris type by the assuption of quadrant negative dependence, consuers who purchase insurance have lower average ris than the entire population. Proposition 5 represents a striing difference fro the result reported in Proposition 1 for the case of perfectly copetitive aret, where we show that negative correlation property will not eerge under any joint distribution of M and Λ. To better explain the intuition why aret structure plays such an iportant role in whether or not negative dependence between M and Λ can lead to negative correlation property in equilibriu, it is useful to further exaine the difference, when gets large, between the copetitive equilibriu price p and the onopolistic price p. 26 In Lea 3 of the Appendix, we show that under onopoly, p / converges to λ when gets large. Moreover, it is straightforward to show that p E M] + λ when is large enough. 27 The difference between copetitive and onopolistic aret structure can be understood as follows. As gets larger, it becoes less costly for the insurance firs to offer insurance due to 26 We use p and p to indicate the copetitive equilibriu price and the profit-axiizing price respectively when the relative iportance paraeter of the ris preference in WTP is. p is forally defined in the Appendix in To see this, suppose > := E M] ] /λ and p > E M] + λ. Then fir s expected profit is πp ; = θ Bp p ] dh, λ > θ Bp violating the zero profit condition required for the copetitive equilibriu. E M] + ] λ dh, λ = 0, 18

20 λ + λ = p + λ = p * λ E M ] Figure 2: Perfect Copetition versus Monopoly: the Case of High the increase in the WTP for insurance for consuers of all ris types. This is true for both the copetitive and the onopolistic aret structure. However, in a perfectly copetitive aret, the copetitive pressure will force insurance copanies to reduce prices and cover ore consuers, leading to a low price level in equilibriu. Specifically, the copetitive equilibriu price is lower than E M] + λ when is sufficiently large. As a result, only consuers with low ris lower than the unconditional expectation and low degree of ris aversion choose to opt out the lower dashed line in Figure 2. This iplies directly that consuers with no insurance have lower average ris than the entire population. 28 In contrast, a onopolistic fir recognizes that, when M and Λ exhibit strong negative dependence, ost densities concentrate on the diagonal as indicated in Figure 2. To axiize profit, a onopolist will choose a higher price relative to the copetitive equilibriu price p so as to exclude the higher ris consuers the upper solid line in Figure 2. Coparative Statics with Respect to the Degree of Negative Dependence Proposition 5 is a liiting result under onopoly as the role of preference as a deterinant of the deand for insurance becoes sufficiently iportant. In this subsection, we further paraeterize the nature of the negative dependence between M and Λ, and exaine its ipact on the equilibriu outcoes of the onopolistic aret, including preiu, aret size and the correlation between insurance 28 In the extree case of perfect negative dependence, every consuer will buy insurance for any > 0, hence E M B p ] = E M]. 19