# «Abstract Both Akerlof (1970) and Rothschild and Stiglitz (1976) show that insurance markets may “unravel”. This memo clariﬁes the distinction ...»

Unraveling versus Unraveling: A Memo on Competitive

Equilibriums and Trade in Insurance Markets

Nathaniel Hendren

January, 2012

Abstract

Both Akerlof (1970) and Rothschild and Stiglitz (1976) show that insurance markets may “unravel”. This memo clariﬁes the distinction between these two notions of unraveling in the context

of a binary loss model of insurance. We show that the two concepts are mutually exclusive occurrences. Moreover, we provide a regularity condition under which the two concepts are exhaustive of the set of possible occurrences in the model. Akerlof unraveling characterizes when there are no gains to trade; Rothschild and Stiglitz unraveling shows that the standard notion of competition (pure strategy Nash equilibrium) is inadequate to describe the workings of insurance markets when there are gains to trade.

1 Introduction Akerlof [1970] and Rothschild and Stiglitz [1976] have contributed greatly to the understanding of the potential problems posed by private information on the workings of insurance markets. Akerlof [1970] shows how private information can lead to an equilibrium of market unraveling, so that the only unique equilibrium is one in which only the worst quality good (i.e. the “lemons”) are traded. Rothschild and Stiglitz [1976] show that private information can lead to an unraveling of market equilibrium, in which no (pure strategy) competitive equilibrium exists because insurance companies have the incentive to modify their contracts to cream skim the lower-risk agents from other ﬁrms.

While the term unraveling has been used to describe both of these phenomena, the distinction between these two concepts is often unclear, arguably a result of each paper’s diﬀerent approach to modeling the environment. Akerlof [1970] works in the context of a “supply and demand” environment with a ﬁxed contract, whereas Rothschild and Stiglitz [1976] work in the context of endogenous contracts in a stylized environment with only two types.

This memo develops a generalized binary loss insurance model that incorporates the forces highlighted in both Akerlof [1970] and Rothschild and Stiglitz [1976]. Using this uniﬁed model, we show that the equilibrium of market unraveling (in Akerlof) is a mutually exclusive occurrence from the unraveling of market equilibrium (in Rothschild and Stiglitz). Moreover, under the regularity condition that the type distribution has full support, one of these two events must occur: either there is a Competitive (Nash) Equilibrium of no trade (Akerlof unraveling) or a Competitive (Nash) Equilibrium does not exist (Rothschild and Stiglitz unraveling). Thus, not only are these two concepts of unraveling diﬀerent, but they are mutually exclusive and, in a generic sense, exhaustive of the potential occurrences in an insurance market with private information.

The intuition is straightforward. When the type distribution has full support, trade requires crosssubsidization of types. If some low-risk agent is willing to cross-subsidize higher-risk agents, then the equilibrium will unravel a la Rothschild and Stiglitz [1976]. Competitive (Nash) equilibriums cannot sustain cross-subsidization and break down when agents want to provide it. In contrast, if no agents are willing to cross-subsidize the worse risks in the population, then there exists a unique Nash equilibrium at the endowment: no one on the margin is willing to pay the average cost of worse risks, and any potential contract (or menu of contracts) unravels a la Akerlof [1970].

The next section develops a model that incorporates the intuition of both Akerlof [1970] and Rothschild and Stiglitz [1976] in the context of a binary loss model of insurance with an arbitrary distribution of types.

2 Model Agents have wealth w and face a potential loss of size l which occurs with probability p, which is distributed in the population according to the c.d.f. F (p) with support Ψ. In contrast to Rothschild and Stiglitz [1976], we do not impose any restrictions on F (p). It may be continuous, discrete, or mixed. We let P denote the random variable with c.d.f. F (p), so that realizations of P are denoted with lower-case p. Agents of type p have vNM preferences given by

** pu (cL ) + (1 − p) u (cN L )**

where u is increasing and strictly concave, cL (cN L ) is consumption in the event of (no) loss. We deﬁne an allocation to be a set of consumption bundles, cL and cN L, for each type p ∈ Ψ, A = {cL (p), cN L (p)}p∈Ψ.

We assume there exists a large set of risk-neutral insurance companies, J, which each can oﬀer cj (p), cj L (p) to maximize expected proﬁts1. Following Rothschild menus of contracts Aj = L N p∈Ψ and Stiglitz [1976], we deﬁne a Competitive Nash Equilibrium as an equilibrium of a two stage game.

In the ﬁrst stage, insurance companies oﬀer contract menus, Aj. In the second stage, agents observe the total set of consumption bundles oﬀered in the market, AU = ∪j∈J Aj, and choose the bundle which maximizes their utility. The outcome of this game can be described as an allocation which satisﬁes the following constraints.

Deﬁnition 1. An allocation A = {cL (p), cN L (p)}p∈Ψ is a Competitive Nash Equilibrium if In contrast to Rothschild and Stiglitz [1976], we allow the insurance companies to oﬀer menus of consumption bundles, consistent with the real-world observation that insurance companies oﬀer applicants menus of premiums and deductibles.

The ﬁrst three constraints require that a Competitive Nash Equilibrium must yield non-negative proﬁts, must be incentive compatible, and must be individually rational. The last constraint rules out the existence of proﬁtable deviations by insurance companies. For A to be a competitive equilibrium, there cannot exist another allocation that an insurance company could oﬀer and make positive proﬁts ˆ on the (sub)set of people who would select the new allocation (given by D A ).

2.1 Mutually Exclusive Occurrences We ﬁrst show that, in this model, the insurance market has the potential to unravel in the sense of Akerlof [1970].

allocation other than A satisfying incentive compatibility, individual rationality, and non-negative proﬁts, contradicting the no-trade theorem of Hendren [2011].

The market unravels a la Akerlof [1970] if and only if no one is willing to pay the pooled cost of worse risks in order to obtain some insurance. This is precisely the logic of Akerlof [1970] but provided in an environment with an endogenous contract space. When Condition (1) holds, no contract or menu of contracts can be traded because they would not deliver positive proﬁts given the set of risks that would be attracted to the contract. This is precisely the unraveling intuition provided in Akerlof [1970] in which the demand curve lies everywhere below the average cost curve. Notice that when this no-trade condition holds, the endowment is indeed a Nash equilibrium. Since no one is willing to pay the pooled cost of worse risks to obtain insurance, there exist no proﬁtable deviations for insurance companies to break the endowment as an equilibrium.

Theorem 1 also shows that whenever the no-trade condition holds, there must exist a Competitive Nash Equilibrium. Thus, whenever the market unravels a la Akerlof [1970], the competitive equilibrium cannot unravel a la Rothschild and Stiglitz [1976]. Unraveling in the sense of Akerlof [1970] is a mutually exclusive occurrence from unraveling in the sense of Rothschild and Stiglitz [1976].

2.2 Exhaustive Occurrences We now show that no only are these two notions of unraveling mutually exclusive, but they are also, exhaustive of the possibilities that can occur in model environments when the type distribution satisﬁes the following regularity condition.

**Assumption 1. (Full support near p = 1) F(p) 1 for all p 1**

Assumption 1 assumes that one cannot rule out the chance of risks arbitrarily close to p = 1.

In other words, we assume there does not exist a highest risk type, p, such that p 1.2 With this ¯ ¯ assumption, an insurance company cannot oﬀer any insurance contract other than the endowment without being worried it will be selected by more than one type. Thus, in order to provide insurance, types must be cross-subsidized.

We now show that competitive equilibriums cannot sustain cross-subsidization, an insight initially provided in Rothschild and Stiglitz [1976].

** Lemma 1. (Rothschild and Stiglitz [1976]) Suppose A is a Competitive Nash Equilibrium.**

Then

Proof. Suppose there exists p such that pcL (p)+(1 − p) cN L (p) w −pl. Then an insurance company ˆ could oﬀer a new allocation, A, which provides the endowment to type p, so that type p now chooses the allocation oﬀered by remaining ﬁrms (essentially dumping type p onto other insurance companies).

Note that this assumption can be satisﬁed with any small amount of mass of types; it is only a condition on the support of the distribution.

Therefore, pcL (p) + (1 − p) cN L (p) ≤ w − pl for all p ∈ Ψ. So, condition (1) implies pcL (p) + (1 − p) cN L (p) = w − pl for all p.

Given Lemma 1, it is straightforward to see that there cannot exist any Competitive Nash Equilibrium other than the endowment since trade requires cross-subsidization toward types near p = 1.

** Theorem 2. Suppose Assumption 1 holds.**

Then, there exists a Competitive Nash Equilibrium if and only if Condition (1) holds.

Proof. Suppose Condition (1) does not hold. Then, clearly there exists a proﬁtable deviation from the endowment. But, Lemma 1 and Assumption 1 ensure that no allocation other than the endowment can be a Competitive Nash Equilibrium.

When Assumption1 holds, trade requires risk types to be willing to enter risk pools which pool exante heterogeneous types. Such ex-ante pooling is not possible in a Competitive Nash Equilibrium. So,

**when the no-trade condition (1) does not hold, there does not exist any Competitive Nash Equilibrium:**

the equilibrium unravels a la Rothschild and Stiglitz [1976].

3 Conclusion This memo uses a generalized binary model of insurance to highlight the distinction between Akerlof’s notion of unraveling, in which an equilibrium exists in which no trade can occur, and Rothschild and Stiglitz’ notion of unraveling, in which a standard notion of competitive equilibrium (pure strategy Nash) cannot exist. In the latter case, there are (Pareto) gains to trade; but when the type distribution has full support near p = 1, the realization of these gains to trade require cross-subsidization of types.

Such cross-subsidization cannot be sustained under the canonical notion of competition.3 In sum, Akerlof unraveling shows when private information can lead to the absence of trade in insurance markets. Rothschild and Stiglitz unraveling shows that the canonical model of competition (Nash equilibrium) is inadequate to describe the behavior of insurance companies in settings where there are potential gains to trade.

References G Akerlof. The market for lemons: Qualitative uncertainty and the market mechanism. Quarterly journal of economics, 84(3):488–500, 1970. 1, 2.1, 2.1 N Hendren. Private information and insurance rejections. Working Paper, 2011. 2.1 H Miyazaki. The rat race and internal labor markets. The Bell Journal of Economics, 8(2):394–418, 1977. 3 In the modiﬁed models of competition, proposed by Miyazaki [1977], Wilson [1977], or Spence [1978], such gains to trade will be realized in equilibrium.

M Rothschild and J Stiglitz. Equilibrium in competitive insurance markets: An essay on the economics of imperfect information. The Quarterly Journal of Economics, pages 629–649, 1976. 1, 2, 1, 2.1, 2.2, 1, 2.2 A.M Spence. Product diﬀerentiation and performance in insurance markets. Journal of Public Economics, 10(3):427–447, 1978. 3 C Wilson. A model of insurance markets with incomplete information. Journal of Economic Theory,