Multidimensional private information, market structure, and insurance markets

• Date: 01 September 2018
• Published date: 01 September 2018
• Author: Zenan Wu,Hanming Fang
• DOI: http://doi.org/10.1111/1756-2171.12251

RAND Journal of Economics

Vol.49, No. 3, Fall 2018

pp. 751–787

Multidimensional private information,

market structure, and insurance markets

Hanming Fang∗

and

Zenan Wu∗∗

We investigate whether selection based on multidimensional private information in risks and

risk preferences can, under different market structures, result in a negative correlation between

insurance coverage and ex post realization of risk. We show that, under perfect competition, se-

lection based on multidimensional private information does not result in the negative correlation

property, unless there is a sufficiently high loading factor. However, it is possible to generate

the negative correlation property under monopoly when risk and risk preference types are suffi-

ciently negative dependent. We also clarify the connections between important concepts such as

adverse/advantageous selection and positive/negative correlation property.

1. Introduction

The classic asymmetric information models of insurance pioneered by Arrow (1963), Pauly

(1974), Rothschild and Stiglitz (1976), and Wilson (1977) assume that potential insurance buyers

have one-dimensional private information regarding their risk type. These models predict a

positive correlation between insurance coverage and ex post realizations of losses. The reason

is ex ante adverse selection, namely, that the “bad risks” (i.e., those relatively likely to suffer

a loss) have a higher willingness to pay for insurance; and allowing for ex post moral hazard

only strengthens the positive correlation between coverage and ex post losses. This “positive

correlation property” of the classic asymmetric information models forms the basis for empirical

tests of asymmetric information in several recent articles (see Chiappori and Salani´

e, 2000).

However, the results from a growing empirical literature testing for the correlation between

insurance coverage and ex post realization of risks are mixed and vary by market. In an auto

insurance market, Chiappori and Salani´

e (2000) find that the accident rate for young French

∗University of Pennsylvaniaand NBER; hanming.fang@econ.upenn.edu.

∗∗Peking University; zenan@pku.edu.cn.

Weare grateful to Mark Armstrong (the Editor) and two anonymous referees for very detailed comments that significantly

improved the article. We would like to thank Eduardo Azevedo, David de Meza, Daniel Gottlieb, Ben Lester, Stephen

Morris, Yeneng Sun, Venky Venkateswaran, Glen Weyl, and seminar/conference participants at National University of

Singapore, Monash University,Rice University, NBER Insurance WorkingGroup Conference (2017), and Asian Meeting

of the Econometric Society (2017), for helpful discussions, suggestions, and comments. Part of Fang’s research on this

project is funded by the generous financial support from NSF grant no. SES-0844845. Wu thanks the School of Economics

at Peking University for research support. All remaining errors are our own.

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drivers who choose comprehensive automobile insurance is not statistically different from those

opting for the legal minimum coverage,after controlling for consumers’ characteristics observable

to the automobile insurers. In contrast, Cohen (2005), using data from an online Israeli insurer,

finds that new auto insurance customers choosing a low deductible contract tend to have more

accidents, leading to higher total losses for the insurer.1In the life insurance market, Cawley and

Philipson (1999) find that the mortality rate of US males who purchase life insurance is below that

of the uninsured, even when controlling for many factors such as income that may be correlated

with life expectancy.2For the long term care (LTC) insurance market, Finkelstein and McGarry

(2006), using panel data from a sample of Americans born before 1923 (the AHEAD study), find

no statistically significant correlation between their LTCcoverage in 1995 and their use of nursing

home care between 1995–2000, even after controlling for the insurers’ assessment of a person’s

risk type. Moreover, when Finkelstein and McGarry (2006) use whether respondents undertake

various types of preventive healthcare as a proxy for risk aversion, they find that people who

are more risk averse by this measure are both more likely to own LTC insurance and less likely

to enter a nursing home. In an annuity insurance market, Finkelstein and Poterba (2004) find

systematic relationships between the ex post mortality and the annuity characteristics, such as the

timing of payments and the possibility of payments to the annuitants’ estate, but they do not find

evidence of substantive mortality differences by annuity size. For the Medigap insurance market,

Fang, Keane, and Silverman (2008) find that, conditional on controls for Medigap prices, those

with Medigap spend on average $4000 less on medical care than those without, providinga strong

evidence for the negative correlation between Medigap purchase and ex post realization of risk.

These empirical findings fueled an emerging literature on the possibility that multidimen-

sional private information may lead to what has been called “advantageous selection.” 3The

formal theoretical literature is sparse. de Meza and Webb (2001) postulate a model in which

individuals differ in their risk preferences, which they refer to as “timid” and “bold” types. They

assume that more timid types may lowertheir risk exposure through increased insurance purchase

and greater precautionary effort to reduce risks. They show that, in the presence of administra-

tive costs in processing claims and issuing policies, there exists a pure-strategy, partial pooling,

subgame-perfect Nash equilibrium in the insurance market that exhibits the negative correla-

tion property. Thus, failure to condition on risk aversion may then mask the positive correlation

between insurance coverage and ex post risk predicted by one-dimensional models. Following

de Meza and Webb(2001), the existing literature points to risk preferences as the primary suspect

behind advantageous selection. In general, however, any private information could function as

a source of advantageous selection if it is positively correlated with insurance coverage and at

the same time negatively correlated with risk. Finkelstein and McGarry (2006) argue that their

findings on the LTCinsurance market is consistent with multidimensional private information and

advantageous selection based on risk aversion. In fact, their findings suggest that, on net, adverse

selection based on risk and advantageous selection based on risk aversion roughly cancel out in

the LTC insurance market. Fang, Keane, and Silverman (2008) find that, for Medigap insurance

market, risk preferences do not appear as a source of advantageous selection, but cognitiveability

is particularly important.

However, to the best of our knowledge, the precise conditions under whichwhether selection

based on multidimensional private information may generate in equilibrium a positiveor negative

1Others have examined the evidence of asymmetric information in the choice of insurance contracts such as

deductibles and copayments, etc. For example, Puelz and Snow (1994) study automobile collision insurance and argue

that, in an adverse selection equilibrium, individuals with lower risk 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 from an automobile insurer in Georgia. However, see Chiappori and

Salani´

e (2000) and Dionne, Gouri´

eroux, and Vanasse(2001) for critiques of the Puelz and Snow study.

2See He (2009) for a reexamination of the evidence.

3The first description of this phenomenon in the economics literature appears to be Hemenway (1990), who used

the term “propitious selection.”

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FANG AND WU / 753

correlation between insurance purchase and ex post realization of risk is still unknown. Most

of the existing articles that invoked the possibility of multidimensional private information as

a possible explanation for the empirical findings discussed above rely on partial equilibrium

intuition (much in the spirit of Hemenway, 1990). An important exception is Chiappori, Jullien,

Salani´

e, and Salani´

e (2006, henceforth, CJSS), which argues that in a competitive insurance

market, the positive correlation property is a general implication of insurance models with

asymmetric information, even when the private information is multidimensional in risks and

risk preferences. The key assumptions are consumer 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.4CJSS’s approach is general and elegant, and they

prove their results using revealed preference and the NIP condition. However, the nonincreasing

profit condition is not a primitive condition; thus, whether it holds in equilibrium in environments

where the market may not be competitive and where loading costs for offering insurance exist is

still an open question.

The goal of this article is to help fill in this gap. Wepresent a simple model of insurance market

where consumers havemultidimensional private information in risk and risk preference types, and

investigate whether selection based on multidimensional private information can, under different

market structures, result in negative correlation in equilibrium between insurance coverage and

ex post realization of risk. Weshow that if the insurance market is perfectly competitive, selection

based on multidimensional privateinfor mation does not generate the negativecorrelation proper ty

in equilibrium unless there is a sufficiently high loading factor,possibly because of administrative

or marketing costs. If the insurance market is monopolistic, however, we show that it is possible

to generate the negative correlation property in equilibrium when consumers’ risk type and risk

preference type are sufficiently negative dependent, a notion we formalize using the concept of

copula. It should be noted that the fully specified model considered in our article is not as general

as that in CJSS; our contribution is to state the connections between the primitives of the model

(including multidmensional private information and market structure) and the positive or negative

correlation property in a transparent way.

The remainder of the article is structured as follows. In Section 2, we provide a detailed

discussion of the related literature. In Section 3, we describe our model environment in which

consumers are heterogeneous in both risk and risk preference types. In Section 4, we consider

the perfectly competitive market structure. In Section 5, we analyze the monopolistic market.5

In Section 6, we clarify the confusions in this growing literature about the connections between

some of the important concepts such as adverse/advantageous selection and positive/negative

correlation property. In Section 7, we partially endogenizethe 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 mild generalizations of the assumptions imposed in Section 3. In Section 8, we summarize our

main findings and suggest directions for future research. All proofs are relegated to an Appendix.

2. Related literature

To the extent that our article investigates on whether the positive correlation property is

robust to environments with multidimensional consumer heterogeneity, it is most related to CJSS

(2006) and de Meza and Webb (2017). CJSS argue that, as long as consumers 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 multidimensional private information. This conclusion is similar to our results for the

competitive insurance market presented in Propositions 1–2 and Proposition 7. Their results are

proved using the revealed preference argumentimplied by the hypothesized consumer rationality

4Wewill discuss the connection between our results and theirs in Section 4.

5In an online web Appendix, we showthat our results for the monopolistic market structure can be generalized to

a version of an imperfectly competitive market structure.

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