Introducing Heterogeneity in the Rothschild-stiglitz Model

In their seminal work, Rothschild and Stiglitz (1976) have shown that in competitive insurance markets, under asymmetric information, pooling contracts cannot exist in equilibrium, firms make zero profit, and, under some circumstances, equilibrium does not exist. In the present work, the model is extended by introducing unobservable wealth in addition to the differing risks. The study shows that if the differences in wealth are small, different wealth types are pooled while different risks are separated. For large wealth differences, partial risk pooling contracts, in which one type chooses different contracts in equilibrium, are feasible. Furthermore, equilibria with profitmaking contracts can exist. Complete risk pooling contracts can occur only under very restrictive assumptions. The effect of the extra dimension of asymmetric information on the nonexistence problem is ambiguous. INTRODUCTION Since the seminal work by Rothschild and Stiglitz (1976), it has become clear that competitive insurance markets under asymmetric information display features not known from the conventional competitive analysis. Contracts specify both quantity and price, and it may happen that equilibrium does not exist. The nonexistence problem has led to a lengthy debate in the literature. Solutions proposed are other equilibrium concepts (Wilson, 1977; Miyazaki, 1977; Spence, 1978), extensions to multi-stage games (Grossman, 1979; Hellwig, 1987; Asheim and Nilssen, 1996), or the introduction of mixed strategies (Dasgupta and Maskin, 1986). Although the equilibrium nonexistence problem has been the main focus of research, some other results by Rothschild and Stiglitz deserve attention. First, they showed that in a competitive environment, pooling contracts cannot exist in equilibrium, and second (maybe less surprisingly), they demonstrated that firms make zero profits. In other equilibrium concepts, the existence of pooling contracts is possible; however, the no-profit result always holds. Achim Wambach is a member of the Department of Economics, University of Munich, Germany. He thanks Klaus Schmidt, Ray Rees, Richard MacMinn, one anonymous referee, and participants of the 1997 Seminar of the European Group of Risk and Insurance Economists for helpful comments and suggestions. 580 THE JOURNAL OF RISK AND INSURANCE As was already argued by Spence (1978, p. 427), “not only may individuals differ in the expected cost they impose upon the insurer, they may also differ in their preferences with respect to insurance coverage.” Following up on this remark, the author considers individuals who differ in risk aversion in addition to their risks. There are two dimensions of asymmetric information—the insurer observes neither the risk nor the degree of risk aversion of the individual. The author models the different degrees of risk aversion by assuming that individuals have decreasing absolute risk aversion and different wealth levels. Other studies on multidimensional adverse selection problems in the context of insurance markets include the ones by Fluet and Pannequin (1997) and Landsberger and Meilijson (1996). The first authors model two types of individuals with multiple risks. Their main result is that bundling insurance contracts can be efficiency enhancing. Landsberger and Meilijson analyze the case in which two types of individuals differ with respect to their distribution of losses. However, they are mainly concerned with the derivation of assumptions under which a first-best result can be achieved or approximated. In contrast to these studies, this article assumes that there are four different types of individuals: those with high or low risks with either high or low wealth. To the author’s knowledge, this is the first study that analyzes the insurance market with four unobservable types. It derives the following new results. First, in the standard case, where the single crossing property holds in the relevant region of the contract space, different wealth types are pooled, while different risk types are separated. This pooling is the same as in the first-best situation, in which different risks obtain full insurance at their fair premium, which is independent of the wealth. Second, the effect of the additional dimension on the equilibrium nonexistence problem is ambiguous: It may happen that if wealth is observable, no equilibrium would exist, but it does exist for unobservable wealth, and vice versa. Third, if the single crossing property does not hold, then even under perfect competition with free entry an equilibrium may exist in which firms make positive profits.1 Fourth, if the single crossing property does not hold, partial risk pooling, where one type mixes between two contracts and another risk type chooses one of the two contracts, can exist in equilibrium. Fifth, complete risk pooling, where different risk types choose the same contract with a probability of one, exists only under very restrictive assumptions. Generalizing from the results, the author is led to conclude that an equilibrium in a competitive insurance market under multidimensional adverse selection is either with a pooling of types that are also pooled in the first best and at the same time with complete separation of risks, or with a partial risk pooling, where some types with the same risk are separated. The study is structured as follows. The author first presents the model and discusses the slope of the indifference curves as a function of risk and wealth. The study then deals with the standard case, where the single crossing property holds in the relevant region of the contract space, before the violation of the single crossing property is considered. Finally, the results are summarized. 1 This result has been independently shown by Villeneuve (1997) and Smart (2000). Villeneuve uses a model with only two types and very general utility functions. Smart, who also works with two types only, employs a different equilibrium concept. INTRODUCING HETEROGENEITY IN THE ROTHSCHILD-STIGLITZ MODEL 581 THE MODEL Consider an individual who has the probability Q of losing L. Her initial wealth is X , and her utility function displays decreasing absolute risk aversion. The insurer offers a contract that specifies a premium P and an indemnity I that will be paid if a loss occurs. Thus the expected utility of the individual is: , , , , 1 V L P I U P U P L I Q X Q X Q X (1) where U is a concave increasing function that satisfies decreasing absolute risk aversion, i.e., / / 0 d U x U x dx bb b . The literature so far has been concerned with an adverse selection problem on the probability of losses Q . In this study, the discussion will be extended for unobservable X .2 The slope of the indifference curve in (I, P) space is given by: