Non-monotone insurance contracts and their empirical consequences

The goal of this paper is to show the possibility of a non-monotone relation between coverage ans risk which has been considered in the literature of insurance models since the work of Rothschild and Stiglitz (1976). We present an insurance model where the insured agents have heterogeneity in risk aversion and in lenience (a prevention cost parameter). Risk aversion is described by a continuous parameter which is correlated with lenience and for the sake of simplicity, we assume perfect correlation. In the case of positive correlation, the more risk averse agent has higher cosr of prevention leading to a higher demand for coverage. Equivalently, the single crossing property (SCP) is valid and iplies a positive correlation between overage and risk in equilibrium. On the other hand, if the correlation between risk aversion and lenience is negative, not only may the SCP be broken, but also the monotonocity of contracts, i.e., the prediction that high (low) risk averse types choose full (partial) insurance. In both cases riskiness is monotonic in risk aversion, but in the last case there are some coverage levels associated with two different risks (low and high), which implies that the ex-ante (with respect to the risk aversion distribution) correlation between coverage and riskiness may have every sign (even though the ex-post correlation is always positive). Moreover, using another instrument (a proxy for riskiness), we give a testable implication to desentangle single crossing ans non single croosing under an ex-post zero correlation result: the monotonicity of coverage as a function os riskiness. Since by controlling for risk aversion (no asymmetric information), coverage is monotone function of riskiness, this also fives a test for asymmetric information. Finally, we relate this theoretical results to empirical tests in the recent literature, specially the Dionne, Gouruéroux and Vanasse (2001) work. In particular, they found an empirical evidence that seems to be compatible with asymmetric information and non single crossing in our framework. More generally, we build a hidden information model showing how omitted variables (asymmetric information) can bias the sign of the correlation of equilibrium variables conditioning on all observable variables. We show that this may be the case when the omitted variables have a non-monotonic relation with the observable ones. Moreover, because this non-dimensional does not capture this deature. Hence, our main results is to point out the importance of the SPC in testing predictions of the hidden information models.

On the cost of sympathy : a alternative model of insurance markets

The paper provides an alternative model for insurance market with three types of agents: households, providers of a service and insurance companies. Households have uncertainty about future leveIs of income. Providers, if hired by a household, perform a diagnoses and privately learn a signal. For each signal there is a procedure that maximizes the likelihood of the household obtaining the good state of nature. The paper assumes that providers care about their income and also about the likelihood households will obtain the good state of nature (sympathy assumption). This assumption is satisfied if, for example, they care about their reputation or if there are possible litigation costs in case they do not use the appropriate procedure. Finally, insurance companies offer contracts to both providers and households. The paper provides sufficient conditions for the existence of equilibrium and shows that the sympathy assumption 1eads to a 10ss of welfare for the households due to the need to incentive providers to choose the least expensive treatment.

Do insurance defrauders want to be punished?

We analyze a Principal-Agent model of an insurer who faces an adverse selection problem. He is unable to observe if his client has a high risk or a low risk of having an accident. At the underwriting of the contract, the insurer requests the client to declare his risk. After that, the former can costly audit the truthfulness of this announcement. If the audit confirms a false declaration, the insurer is legally allowed to punish the defrauder. We characterize the efRcient contracts when this punishment is bounded from above by a legal restriction. Then, we do some comparative statics on the efRcient contracts and on the agent's utility. The most important result of this paper concerns the legal limit to a defrauder's punishment. We prove that there exists a uni que value of this legal limit that maximizes the expected utility of a high risk type. Facing this particular value of the legal limit to a defrauder's punishment, the insurer will effectively audit a low risk reporto We also show that this particular value increases with the probability of facing a high risk policyholder. Therefore, when this probability is sufRciently high, the nullity of the contract is not enough. From the point of view of a potential defrauder, the law should allow harder sanctions. This is an striking result because the nullity of the contract is a common sanction for this kind of fraud in the USA and in some European countries.

Optimal unemployment insurance for couples

This work analyzes the optimal design of an unemployment insurance program for couples, whose joint search problem in the labor market differ significantly from the problem faced by single agents. We use a version of the sequential search model of the labor market adapted to married agents to compare optimal constant policies for single and married agents, as well as characterize the optimal constant policy when the agency faces single and married agents simultaneously. Our main result is that an agency that gives equal weights to single and married agents will want to give equal utility promises to both types of agents and spend more on the single agent.