# «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, 2014

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. I show that the two concepts are mutually exclusive occurrences.

Moreover, I 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 or asset (e.g. a used car), whereas Rothschild and Stiglitz (1976) work in the context of endogenous contracts in a stylized environment with only two types (e.g. high and low 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, I 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 either (a) contains a continuous interval or (b) includes p = 1, 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 generically exhaustive of the potential occurrences in an insurance market with private information.

The mutual exclusivity result is more or less obvious in the canonical two-type binary loss model.

The market unravels a la Rothschild and Stiglitz when the low type has an incentive to cross-subsidize the high type in order to obtain a more preferred allocation. This willingness of the good risk to subsidize the bad risk is precisely what ensures the market will not unravel a la Akerlof. Conversely, if the market unravels a la Akerlof, then the good risk is not willing to subsidize the bad risk, which implies an absence of the forces that drive non-existence in Rothschild and Stiglitz.

The intuition for the exhaustive result is also straightforward, but perhaps more diﬃcult to see in the context of the stylized two-type model. When the support of the type distribution either (a) contains an interval or (b) contains the point p = 1, then trade necessarily involves cross-subsidization of types.1 But, Rothschild and Stiglitz (1976) show that a competitive (Nash) equilibrium cannot sustain such cross-subsidization. Hence, if agents are willing to provide trade then the market unravels a la Rothschild and Stiglitz. 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 logic can be seen in the canonical two-types case. Here, the regularity condition requires one to assume that the bad risk will experience the loss with certainty. The only way for the low type (good risk) to obtain an allocation other than her endowment is to subsidize the high type (bad risk) away from her endowment. If the low type is willing to do so, the equilibrium unravels a la Rothschild and Stiglitz. If the low type is unwilling to do so, the equilibrium unravels a la Akerlof.

In the two type model, the assumption that the bad risk experiences the loss with certainty is clearly restrictive. However, for more general type distributions beyond the two-type case the regularity condition is quite weak. Any distribution can be approximated quite well by distributions that have continuously distributed regions or by distributions with an arbitrarily small amount of mass at p = 1. In this sense, the existence of pure strategy competitive equilibria of the type found by Rothschild and Stiglitz (1976) that yield outcomes other than the endowment is a knife-edge result.

This highlights the importance of recent and future work to aid in our understanding of how best to model competition in insurance markets.

As discussed below, Riley (1979) shows this is true in the case when the support contains an interval; I show below this is also the case when the support is discrete but includes the point p = 1.

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.2 In contrast to Rothschild and Stiglitz (1976), I do not impose any restrictions on F (p).3 It may be continuous, discrete, or mixed. I 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. I 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.

I assume there exists a large set of risk-neutral insurance companies, J, which each can oﬀer cj (p), cj L (p) to maximize expected proﬁts4. Following Rothschild menus of contracts Aj = L N p∈Ψ and Stiglitz (1976), I 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

3. A is individually rational pu (cL (p)) + (1 − p) u (cN L (p)) ≥ pu (w − l) + (1 − p) u (w) ∀p ∈ Ψ The model is adapted from Hendren (ming), which derives the no-trade condition analogue of Akerlof in the binary loss environment but does not provide any discussion of competitive equilibriums.

To my knowledge, Riley (1979) was the ﬁrst paper to discuss this environment with a continuum of types.

In contrast to Rothschild and Stiglitz (1976), I 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 I ﬁrst show that, in this model, the insurance market has the potential to unravel in the sense of Akerlof (1970).

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).

Two-type case To relate to previous literature, it is helpful to illustrate how Theorem 1 works in the canonical two-type model of Rothschild and Stiglitz (1976). So, let Ψ = pL, pH with pH pL denote the type space and let λ denote the fraction of types pH. When pH 1, Corollary 1 of Hendren (ming) shows that the market cannot unravel a la Akerlof.5 Hence, the mutual exclusivity of Akerlof and Rothschild and Stiglitz holds trivially. But, when pH = 1, the situation is perhaps more interesting. To see this, Figure 1 replicates the canonical Rothschild and Stiglitz (1976) graphs in the case when pH = 1.

The vertical axis is consumption in the event of a loss, cL ; the horizontal axis is consumption in the event of no loss, cN L. Point 1 is the endowment {w − l, w}. Because pH = 1, the horizontal line running through the endowment represents both the indiﬀerence curve of type pH and the actuarially fair line for type pH. Notice that type pH prefers any allocation bundle that lies above this line (intuitively, she cares only about consumption in the event of a loss).

The low type indiﬀerence curve runs through the endowment (point 1) and intersects the 45-degree line parallel to her actuarially fair line. As noted by Rothschild and Stiglitz (1976), the outcomes in this environment depend crucially on the fraction of low versus high types. Figure 1 illustrates the two cases. If there are few pH types (λ is small), then point 2 is a feasible pooling deviation from the endowment. When such a deviation is feasible, unraveling a la Akerlof does not occur: the low type is willing to pay the pooled cost of the worse risks. But, the existence of such a deviation is precisely what breaks the existence of a competitive equilibrium in Rothschild and Stiglitz (1976).

Point 2 involves pooling across types and cannot be a competitive equilibrium. Hence, if the market unravels a la Akerlof, the endowment is the unique competitive equilibrium. If the market unravels a la Rothschild and Stiglitz, there exists implementable allocations other than the endowment and Akerlof’s notion of unraveling does not occur.

As one might gather from Figure 1, when types are arbitrarily close to 1, the only feasible competitive equilibrium is the endowment – i.e. there is no possibility of a pair of separating contracts with the pL -type receiving partial coverage in equilibrium. I now make this point in the more general setting that does not require any mass of types at pH = 1.

If pH 1, then equation (1) would be violated at pH = 1 by the assumption of strict concavity of u.

2.2 Exhaustive Occurrences I now show that not 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. Either (a) there exists a b such that [a, b] ⊂ Ψ or (b) 1 ∈ Ψ (i.e. F(p) 1 for all p 1).

Assumption 1 assumes that the support of the type distribution includes either (a) a continuous interval or (b) the point p = 1. Note any distribution can be approximated arbitrarily closely by distributions satisfying this regularity condition.

I 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. See Rothschild and Stiglitz (1976) for a full discussion. Clearly, competition requires zero proﬁts on any consumption bundle. Hence, it suﬃces to show that no allocation can pool types into the same consumption bundle other than the endowment. Suppose multiple types are allocated to the same consumption bundle (distinct from the endowment). Then, an insurance company could oﬀer a ˆ new allocation, A, arbitrarily close to the current allocation but that is only preferred by the lowest p in the pool. Hence, this allocation will provide strictly positive proﬁts and will render the original consumption bundle unproﬁtable, thereby breaking the Nash equilibrium with pooling. Therefore, pcL (p) + (1 − p) cN L (p) = w − pl for all p.