Asymmetric information in insurance: general testable implications.

Several recent articles on empirical contract theory and insurance have tested for a positive correlation between coverage and ex post risk, as predicted by standard models of pure adverse selection or pure moral hazard. We show here that the positive correlation property can be extended to general setups: competitive insurance markets and cases where risk aversion is public. We test our results on a French dataset. Our tests confirm that the estimated correlation is positive; they also suggest the presence of market power.

1. Introduction

* Asymmetric information is present on all markets. Whatever the product or service sold, the seller almost never knows the buyer's preferences, nor the maximum price she would be willing to pay to acquire it. Similarly, the buyer is in general unlikely to have much information about the seller's production technology or marginal costs. Most of the time, however, this asymmetry is irrelevant. In a perfectly competitive setting, the seller would not benefit from a detailed knowledge of the buyer's willingness to pay, because he has to charge the competitive price; the buyer needs no information about the technology, since again all the information she needs is contained in the price. Hence, asymmetric information is in general both paramount and inconsequential.

A common feature of the examples above is that the value of the hidden information is private (in the sense that the payoff of the uninformed party does not depend on it for a given contract). Fagart (1996) proves under weak assumptions that competition in the private-values case always leads to an equilibrium, which is moreover efficient. For want of a better term, we will call such a case "irrelevant asymmetric information" in the present article. The main innovation of the literature on asymmetric information, as pioneered by Akerlof (1970), Rothschild and Stiglitz (1976), and many others, was to exhibit cases in which, on the contrary, asymmetric information was indeed relevant--and actually had important consequences for the existence and efficiency of competitive equilibrium. The key property driving this conclusion is the presence of "common values," in the form of a link between an agent's hidden information and the other party's payoff. In the market for lemons for instance, the buyer's payoff depends on the quality of the car, which is known only by the seller. Similarly, an insurer's profit depends on the risk of the insurees who buy contracts from him.

When considering empirical applications of such models, the previous remarks have important consequences. One is that evidence of information asymmetries, while relatively easy to produce, are often of little interest unless the asymmetries are of the relevant type. To give only one example, agents are often faced with menus of contracts. Menus of contracts are indeed suggestive of asymmetric information. Most of the time, however, this asymmetry is irrelevant. New cars are offered in different colors, which indeed reflects the seller's ignorance about the buyer's taste. Still, market equilibrium will typically exist and be efficient as usual, as the buyer's taste does not directly affect the seller's payoff. Different levels of insurance coverage may be proposed to insurees, reflecting asymmetric information about risk aversion. Insofar as differences in risk aversion have no impact on the insurer's profit, however, the Akerlof-Rothschild-Stiglitz conclusions do not apply, and standard analysis is still valid. This simply reflects the fact that in a competitive setting, the insuree's true risk matters to the insurer, even conditional on the insuree's contract choice, while risk aversion does not. The former is a case of common values, and the latter a case of private values.

Clearly, one should primarily be interested in testing for asymmetric information in the "relevant" case. The main purpose of the article is precisely to propose robust empirical tests of relevant information asymmetries. Throughout, we concentrate on the particular case of insurance contracts, both because the main theoretical contributions to competition under adverse selection (starting with Rothschild and Stiglitz's seminal article) used this framework, and because a large fraction of existing empirical literature deals with insurance contracts. However, our conclusions are general, and the methodology developed here could be useful in other cases.

In the literature on insurance, the general notions just sketched lead to a well-known property on which recent empirical work has largely focused. (1) Under both moral hazard and "relevant" adverse selection, one should observe a positive correlation (conditional on observables) between risk and coverage: if different insurance contracts are actually sold to observationally identical agents, then the frequency of accidents among the subscribers of a contract should increase with the coverage it offers. In the Rothschild and Stiglitz (1976) model of competition under adverse selection, where riskiness is an exogenous and unobservable characteristic of agents, the correlation stems from the fact that "high-risk" agents are ready to pay more than "low-risk" ones for additional coverage, and will therefore choose contracts with higher coverage. Under pure moral hazard, as in Arnott and Stiglitz (1988), an opposite causality generates the same correlation: an agent who, for any unspecified (and exogenous) reason, switches to a contract with greater coverage makes less effort and thus becomes riskier.

The "positive correlation" prediction is appealing, but its robustness may however be questioned--a standard problem facing any empirical work on the topic. Theoretical models of asymmetric information typically use oversimplified frameworks, which can hardly be directly transposed to real-life situations. Rothschild and Stiglitz's model assumes that accident probabilities are exogenous (which rules out moral hazard), that only one level of loss is possible, and more strikingly that agents have identical preferences which are moreover perfectly known to the insurer. The theoretical justification of these restrictions is straightforward: analyzing a model of "pure," one-dimensional adverse selection is an indispensable first step. But their empirical relevance is dubious, to say the least. In real life, moral hazard can hardly be discarded a priori (and interacts with adverse selection in a nontrivial way, as precaution depends on risk and preferences (2)); losses are continuous variables, often ranging from small amounts to hundreds of thousands of dollars; and preference heterogeneity is paramount and largely unobserved.

The first part of our article is devoted to a theoretical analysis of this issue. We show that the positive correlation property derived from Rothschild and Stiglitz extends to much more general models, as already conjectured by Chiappori and Salanie (2000), although its form and robustness vary with the type of competition at stake. Specifically, we extend the property in two directions. First, we consider the case of competitive markets, and we show that relevant asymmetric information (with any combination of adverse selection and moral hazard that generates common values) indeed implies a positive correlation between risk and coverage, for suitably defined such notions. This result is a direct extension of Rothschild and Stiglitz's initial idea to a very general framework (entailing heterogeneous preferences, multiple level of losses, multidimensional adverse selection plus possibly moral hazard, and even nonexpected utility). Second, we study the case of imperfect competition, and we underline the key role of the agent's risk aversion. If it is public information, then some form of positive correlation must hold. In particular, with only one level of loss and expected utility, contracts with higher coverage must exhibit a larger frequency of accidents. Conversely, if risk aversion is private information, the property does not necessarily hold: this was shown in Jullien, Salanie, and Salanie (2007). Risk aversion thus is a key parameter whose informational status drives the testable implications of simple models in the presence of market power.

In the second part of the article, we illustrate the theoretical analysis by testing the predictions it generates on a dataset collected by a large French car insurer. We first test the general relevance of our setting and, in particular, of the assumption that agents correctly assess their accident probability. Our test uses a revealed-preference argument that is robust to any assumption on the information structure or the nature of competition. We find that the data strongly corroborate the predicted property, which validates our approach. We then test for the positive correlation property, and we find evidence of a positive (generalized) correlation. A closer examination of the data suggests that the insurer's profits are probably higher for contracts with a higher coverage, contrary to the predictions of competitive models. This suggests that more work should be devoted to analyzing imperfectly competitive models of insurance markets.

Section 2 builds a general model of insurance under asymmetric information. In Section 3 we apply a revealed-preference argument to obtain a first testable implication that relates the premium differential to expected indemnities. Section 4 analyzes the robust version of the correlation property; we show that it holds both when competition drives profits to zero and when risk aversion is public information. Section 5 tests the properties derived in Sections 3 and 4. Section 6 concludes.

2. The model