A general model of insurance under adverse selection

Summary. This paper considers optimal insurance schemes in a principal-agent multi-dimensional environment in which two types of risk averse agents differ in both risk and attitude to risk. Risk corresponds to any pair of distribution functions (not necessarily ordered by any of the usual dominance relations) and attitudes to risk are represented by any pair of non-decreasing and concave utility functions (not necessarily ordered by risk aversion). Results obtained in one-dimensional models that considered these effects separately and under more restricted conditions, are preserved in the more general set-up, but some of the questions we study can only be posed in the more general framework. The main results obtained for optimal insurance schemes are: (i) Insurance schemes preserve the order of certainty equivalents; consequently, the latter constitute a one-dimensional representation of types. (ii) Agents with the lower certainty equivalent are assigned full insurance. Partial insurance assigned to the others may entail randomization. (iii) Partially insured positions are an increasing function of the ratios of the probabilities that the two types assign to the uninsured positions. Most of these properties are preserved when, due to competition or other reasons, the insured certainty equivalents can not be set below pre-determined levels. Received: January 13, 1998; revised version: October 10, 1998     

Production and insurance under regret aversion

This paper examines the behavior of a regret-averse producer facing revenue risk. To insure against the revenue risk, the producer can purchase a coinsurance contract with an endogenously chosen coinsurance rate. Regret-averse preferences are characterized by a utility function that includes disutility from having chosen ex-post suboptimal alternatives. We show that the regret-averse producer never fully insures against the revenue risk even though the coinsurance contract is actuarially fair. When the producer is sufficiently regret averse and the loss probability is high, we further show that the regret-averse producer chooses not to purchase the actuarially fair coinsurance contract. Even when purchasing the actuarially fair coinsurance contract is optimal, we derive sufficient conditions under which the regret-averse producer reduces the optimal output level as compared to that without the coinsurance contract. These results are distinct from those under pure risk aversion, thereby making the consideration of regret aversion crucial.     

Optimal portfolio positioning under ambiguity

This paper analyzes the optimality of financial portfolios when the investor has a utility with ambiguity aversion. It provides a general result about the optimal portfolio profile under ambiguity, in the Anscombe–Aumann framework, using the Maccheroni et al. (2006) approach which includes Gilboa and Schmeidler's (1989) multiple prior preferences and Hansen and Sargent's (2011) multiplier preferences. The paper then details the CRRA case with an ambiguity index based on relative entropy. Such findings have practical applications in structured portfolio management. Indeed, it is important to take account of uncertainty about the true values of financial parameters when determining the best portfolio profile.     

Intertemporal substitution, risk aversion and ambiguity aversion

Summary. This paper axiomatizes a form of recursive utility on consumption processes that permits a role for ambiguity as well as risk. The model has two prominent special cases: (i) the recursive model of risk preference due to Kreps and Porteus [18]; and (ii) an intertemporal version of multiple-priors utility due to Epstein and Schneider [8]. The generalization presented here permits a three-way separation of intertemporal substitution, risk aversion and ambiguity aversion.Received: 5 August 2003, Revised: 12 March 2004, JEL Classification Numbers: D80, D81, D90.I am grateful to Larry Epstein for his guidance and invaluable advice, and to a referee for helpful comments and suggestions.     

Optimal risk‐sharing under adverse selection and imperfect risk perception

Abstract.  The present paper thoroughly explores second‐best efficient allocations in an insurance economy with adverse selection. We start with a natural extension of the classical model, assuming less than perfect risk perception. We characterize the constraints on efficient redistribution, and we summarize the incidence of incentives on the economy with the notions of weak and strong adverse selection. Finally, we show in what sense improving risk perception enhances welfare.     

Optimal incentive contracts under inequity aversion

We analyze the classic moral hazard problem with the additional assumption that agents are inequity averse. The presence of inequity aversion alters the structure of optimal contracts. When the concern for equity becomes more important, there is convergence towards linear sharing rules. The sufficient statistics result is violated. Depending on the environment, contracts may be either overdetermined, i.e. include non-informative performance measures, or incomplete, i.e. neglect informative performance measures. Finally, our model delivers a simple rationale for team based incentives, implying wage compression.     

Why uncertainty matters: discounting under intertemporal risk aversion and ambiguity

Uncertainty has an almost negligible impact on project value in the standard economic model. I show that a comprehensive evaluation of uncertainty and uncertainty attitude changes this picture fundamentally. The illustration of this result relies on the discount rate, which is the crucial determinant in balancing immediate costs against future benefits, and the single most important determinant of optimal mitigation policies in the integrated assessment of climate change. First, the paper removes an implicit assumption of (intertemporal or intrinsic) risk neutrality from the standard economic model. Second, the paper introduces aversion to non-risk uncertainty (ambiguity). I show a close formal similarity between the model of intertemporal risk aversion, which is a reformulation of the widespread Epstein–Zin–Weil model, and a recent model of smooth ambiguity aversion. I merge the models, achieving a threefold disentanglement between risk aversion, ambiguity aversion, and the propensity to smooth consumption over time.     

An evolutionary analysis of insurance markets with adverse selection

The equilibrium nonexistence problem in Rothschild and Stiglitz's insurance market is reexamined in a dynamic setting. Insurance firms are boundedly rational and offer menus of insurance contracts which are periodically revised: profitable competitors' contracts are imitated and loss-making contracts are withdrawn. Occasionally, a firm experiments by withdrawing or innovating a random set of contracts. We show that Rothschild and Stiglitz's candidate competitive equilibrium contracts constitute the unique long-run market outcome if innovation experiments are restricted to contracts which are sufficiently “similar” to those currently on the market.     

Optimal tax policy and foreign direct investment under ambiguity

We analyze the optimal timing of an irreversible foreign direct investment by a foreign firm and the optimal tax policy by a host country under ambiguity. We derive the optimal GDP level at which the foreign firm switches from exporting to a foreign direct investment. Furthermore, we derive the optimal tax policy by the host country, and analyze the effect of an increase in ambiguity on the optimal tax policy. We show that the host country should reduce the optimal corporate tax rate from the host government’s perspective in response to an increase in ambiguity. Our result is different from the one obtained by Pennings (2005) that shows that an increase in risk induces an increase in the optimal corporate tax rate.     

Unemployment insurance and involuntary unemployment: The case of adverse selection

This paper studies the private and public provision of unemployment insurance in a model where equilibrium unemployment results from the use of the wage as a worker selection device. It is shown that private provision will not occur, essentially because those workers most desired by a firm place least value on such coverage. Respecting the same informational constraints, public provision necessarily gives rise to some distortions in participation decisions, but nonetheless is welfare improving in the aggregate.     

Incomplete insurance and absolute risk aversion

Suppose that a subset of states of nature are not verifiable individually. Given an optimal feasible insurance scheme, the expected utility across a group of unverifiable states is greater (less) than that of a verifiable state, if the degree of absolute risk aversion is decreasing (increasing).     

Genetic adverse selection: Evidence from long-term care insurance and Huntington disease

Individual, personalized genetic information is increasingly available, leading to the possibility of greater adverse selection over time, particularly in individual-payer insurance markets. We use data on individuals at risk for Huntington disease (HD), a degenerative neurological disorder with significant effects on morbidity, to estimate adverse selection in long-term care insurance. We find strong evidence of adverse selection: individuals who carry the HD genetic mutation are up to 5 times as likely as the general population to own long-term care insurance. This finding is supported both by comparing individuals at risk for HD to those in the general population and by comparing across tested individuals in the HD-risk population with and without the HD mutation.     

Direct evidence of risk aversion as a source of advantageous selection in health insurance

Using the self-stated degree of risk aversion regarding health from the GSOEP we find some evidence for risk aversion being a source of advantageous selection. Risk averse men more often procure supplementary insurance for hospital visits despite needing the additional coverage less.     

Moral hazard,adverse selection,and the optimal provision of social insurance

The single-period social insurance model of Diamond and Mirrlees is extended to allow for a diversity of types (in the probability of becoming disabled). When individual type is observable, the utilitarian optimum has both consumption when working and disability benefits increasing with the probability of disability. When type is not observable (adverse selection is present), the optimum is a single ‘pooling’ policy over a wide range of welfare weights which includes the utilitarian case. These results also provide insights into the potential distributional effects of moral hazard and the ways moral hazard and adverse selection problems may interact.     

Minimum standards, insurance regulation and adverse selection: evidence from the Medigap market

This paper examines the implications of minimum standards for insurance markets. I study the imposition of binding minimum standards on the market for voluntary private health insurance for the elderly. The central estimates suggest that the introduction of the standards was associated with an 8 percentage point (25%) decrease in the proportion of the population with coverage in the affected market, with no evidence of substitution toward other, unregulated sources of insurance coverage. To explore possible factors contributing to the impact of the minimum standards, I develop comparative static predictions of the impact of imposing minimum standards in an insurance market with adverse selection. The observed changes in market equilibrium associated with the minimum standards are broadly consistent with these predictions, providing evidence of the existence of adverse selection in this insurance market. More importantly, they suggest that the presence of adverse selection—which in principle may provide an economic rationale for minimum standards—in practice may have exacerbated the declines in insurance coverage associated with the minimum standards.     

Optimal greenhouse gas emissions under various assessments of climate change ambiguity

The probabilities associated with global warming damage are likely to be continuously revised in the light of new information. Such revisions of probability are the defining characteristic of ambiguity, as opposed to risk. This paper examines how climate change ambiguity may affect optimal greenhouse gas emission strategies, via the decision maker's attitude towards anticipated changes of damage probabilities. Two conceptualizations of ambiguity are distinguished, according to the emphasis placed on the ambiguity of priors or on the ambiguity of news, respectively. It is shown that the way in which ambiguity is viewed and the attitude taken towards it have a substantial influence on the optimal emission trajectory.     

The taxation of life annuities under adverse selection

This paper studies how annuities should be taxed in a Mirrlees-type model in the presence of adverse selection and a positive link between income and longevity. The government is able to address the adverse selection problem by implementing a progressive marginal tax rate on annuities. This amounts to subsidizing small annuities (purchased by low incomes) and taxing large annuities (purchased by high incomes). Numerical simulations suggest that the taxation is significant and becomes more pronounced as annuitants get older.     

Optimal contract mechanisms for principal-agent problems with moral hazard and adverse selection

Summary We examine the problem of incentive compatibility and mechanism design for incomplete information principal-agent problems. Allowing for risk aversion on the part of the principal and agent, we show the existence of an optimal, incentive compatible contract selection mechanism for the principal under conditions of moral hazard and adverse selection. Since we assume that the contract set is a function space of state contingent contracts, and that the set of agent types is uncountable, the set of contract selection mechanisms becomes infinite dimensional. Hence, novel existence arguments are required. Our existence result extends those of Grossman and Hart [10] and Page [23] to an infinite dimensional setting with incomplete information.