Rothschild Stiglitz model notes PDF

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Rothschild-Stiglitz model notes We are able to show that not only may a competitive equilibrium not exist, but when equilibria do exist, they may have strange properties. In the insurance market, upon which we focus much of our discussion, sales offers, at least those that survive the competitive process, do not specify a price at which customers can buy all the insurance they want, but instead consist of both a price and a quantity – a particular amount of insurance that the individual can buy at that price. Furthermore, if individuals were willing or able to reveal their information, everybody could be made better off. High-risk individuals cause a negative externality: the low-risk individuals are worse off than they would be in the absence of the high-risk individuals. However, the high-risk individuals are no better off than they would be in the absence of the low-risk individuals. If only the high-risk individuals would admit to their having high sickness probabilities, all individuals would be made better off without anyone being worse off. Thus, competitive equilibria are not Pareto-optimal (in a first-best sense). Set up of the game:  

Players: individuals who buy insurance and companies that sell it Timing: 1. Nature and individual selects probability of being sick, i.e. type 2. Insurance companies offer insurance contracts (uninformed party moves first) 3. Individuals choose which contract to buy

Assumptions: 1. Assume that individuals are identical in all respects except their probability of falling sick. 2. Assume that individuals are risk-averse. 3. Assume that insurance companies are risk-neutral and that they are expected profit maximisers. 4. Assume that insurance companies have financial resources such that they are willing and able to sell any number of contracts that they think will make an expected profit. 5. Assume competitive market with free entry. 6. Assume that customers can buy only one insurance contract. (This is an objectionable assumption because it implies that the seller of insurance specifies both the prices and quantities of insurance purchased. In most competitive markets, sellers determine only price and have no control over the amount their customers buy. Rothschild & Stiglitz defend this assumption, believing it to be more appropriate for their model of the insurance market than traditional price competition. For example, many insurance policies specify that they are not in force if there is another policy.) Information assumptions

Assume that individuals know their risk type, i.e. their probability of falling sick. Assume that insurance companies do not know the individuals’ risk type. Thus, companies cannot discriminate among their potential customers on the basis of their characteristics (since, the individuals are identical in all other respects). It can be possible to force customers to make market choices in such a way that they both reveal their characteristics and make the choices the firm would have wanted them to make had their characteristics been publicly known. This is the self-selection mechanism. Definition of equilibrium Equilibrium in a competitive insurance market is a set of contracts such that, when customers choose contracts to maximise expected utility, (i) no contract in the equilibrium set makes negative expected profits; and (ii) there is no contract outside the equilibrium set that, if offered, will make a non-negative profit. This is a Cournot-Nash type equilibrium notion; each firm assumes that the contracts its competitors offer are independent of its own actions. No pooling equilibrium exists. A separating equilibrium may exist if there are relatively few high-risk individuals. Robustness Alternative information assumptions Suppose individuals do not know their own probability of falling sick, so all assume a mean probability, ´p . But, assume low-risk individuals are less risk-averse on average. Then the low-risk individuals will have a flatter indifference curve than the high-risk individuals. So, no pooling equilibrium exists. There may exist no equilibrium at all and if there does exist an equilibrium, it will entail partial insurance for both groups. Thus, the Rothschild-Stiglitz model’s conclusions do not require that people have particularly good information about their sickness probabilities. All that is required is that individuals with different risk properties differ in some characteristic that can be linked with the purchase of insurance and that, somehow, insurance firms discover this link. Firm behaviour If firms are allowed to offer more than one contract, a separating equilibrium can break down. This is because a firm can enter and offer a profitable contract that is preferable to the low-risk types than the one in the separating equilibrium and a loss-making contract that is preferable to the high-risk types. The profits subsidise the losses so that the company overall breaks even (or maybe even makes profit?). This also shows how a separating equilibrium can be inefficient, because both risk types can be made better off while insurers still break even. But if there are enough high-risk people, then the separating equilibrium can be efficient (because the losses won’t be able to be offset by the profits, which are relatively small). Alternative equilibrium concepts In the basic model, the insurance firm assumes that its actions do not affect the market: the set of policies offered by other firms was independent of its own offering.

It may be the case that firms believe that other firms will respond in some way to any new contract offered. The Wilson equilibrium concept allows for this. A Wilson equilibrium is a set of contracts such that (a) all contracts make non-negative profits and (b) there does not exist a new contract (or set of contracts), which, if offered, makes positive profits even when all contracts that lose money as a result of this entry are withdrawn. If this equilibrium concept is applied to the basic Rothschild-Stiglitz model, then equilibria always exist. The Rothschild-Stiglitz separating equilibrium is also a Wilson equilibrium (check?). When this does not exist, the Wilson equilibrium is the pooling contract that maximises the utility of the low-risk customers. See paper for proof. But, Rothschild & Stiglitz argue that such nonmyopic equilibrium concepts are more appropriate for models of monopoly (or oligopoly) than for models of competition, since it is hard to see why any single firm should take into account the consequences of its offering a new policy in a competitive insurance market (in the absence of collusion or regulation). Potential issue: the insurance market is in fact probably not competitive. How do R&S argue that this isn’t a problem (if they do)???...