We build a model of competitive pooling and show how insurance contracts emerge in equilibrium, designed by the invisible hand of perfect competition. When pools are exclusive, we obtain a unique separating equilibrium. When pools are not exclusive but seniority is recognized, we obtain a diﬀerent unique equilibrium: the pivotal primary-secondary equilibrium. Here reliable and unreliable households take out a common primary insurance up to its maximum limit, and then unreliable households take out further secondary insurance.
Dubey, Pradeep and Geanakoplos, John, "Insurance Contracts Designed by Competitive Pooling" (2001). Cowles Foundation Discussion Papers. 1575.