Competitive Insurance Markets with Unbounded Cost

Abstract

Azevedo and Gottlieb [2017] (AG) define a notion of equilibrium that always exists in the Rothschild and Stiglitz [1976] (RS) model of competitive insurance markets, provided costs are bounded. However, equilibrium predictions are sensitive to assumptions made about the upper bound of cost: introducing an infinitesimal mass of high cost individuals discretely increases all equilibrium prices and reduces coverage for all individuals. We measure model sensitivity to these assumptions by considering sequences of economies with increasing upper bounds of cost, and determining whether the sequence of their equilibria converges. We present sufficient conditions under which AG equilibrium exists when cost is unbounded. For simple insurance markets, we derive a condition which is necessary and sufficient for existence: surplus from insurance must increase faster than linearly with expected cost. This condition is empirically common. If this condition does not hold, a wider range of costs results in market unraveling because all prices increase without bound and, in the limit, an AG equilibrium does not exist. We use these results to show that the equilibrium for an insurance market with an unbounded continuum of types is characterised by a simple differential equation. We also provide examples of non- existence for lemons markets (where a single insurance product is available) with unbounded cost.