Unravelling vs Unravelling: A Memo on Competitive Equilibriums and Trade in Insurance Markets

Introduction

Akerlof¹ and Rothschild and Stiglitz² have contributed greatly to the understanding of the potential problems posed by private information on the workings of insurance markets. Akerlof¹ shows how private information can lead to an equilibrium of market unravelling , so that the only unique equilibrium is one in which only the worst quality good (i.e. the "lemons") are traded. Rothschild and Stiglitz² show that private information can lead to an unravelling 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.

Although the term unravelling 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 modelling the environment. Akerlof¹ works in the context of a "supply and demand" environment with a fixed contract or asset (e.g. a used car), whereas Rothschild and Stiglitz² work in the context of endogenous contracts in a stylised environment with only two types (e.g. high and low types).

This memo develops a generalised binary loss insurance model that incorporates the forces highlighted in both Akerlof¹ and Rothschild and Stiglitz.² Using this unified model, I show that the equilibrium of market unravelling (in Akerlof) is a mutually exclusive occurrence from the unravelling 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 unravelling) or a Competitive (Nash) Equilibrium does not exist (Rothschild and Stiglitz unravelling). Thus, not only are these two concepts of unravelling 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 à la Rothschild and Stiglitz when the low type has an incentive to cross-subsidise the high type in order to obtain a more preferred allocation. This willingness of the good risk to subsidise the bad...