Poverty Traps and Disaster Insurance in a Bi-Level Decision Framework

In this paper, we study mechanisms of poverty traps that can occur after large disaster shocks. Our starting point is a stylized deterministic dynamic model with locally increasing returns to scale possibly generating multiple equilibria paths with finite upper equilibrium. The deterministic dynamics is then overlayed by random dynamics where catastrophic events happen at random points of time. The number of events follows a Poisson process, whereas the proportional capital losses (given a catastrophic event) are beta distributed. In a setup with fixed insurance premium per unit of insured capital, a fraction of the capital might be insured, and this fraction may change after each event. We seek for an optimal strategy with respect to the insured fraction. Falling below the instable equilibrium of the deterministic dynamics introduces the possibility of ending up in a poverty trap after the disaster shocks. We show that the trapping probability (over an infinite time horizon) is equal to one when the stable upper equilibrium of the deterministic dynamics is finite. This is true regardless of the chosen amount of insured capital. Optimization then is done with the expected discounted capital after the next catastrophic event as the objective. Our model may also be useful to assess risk premia and creditworthiness of borrowers when a sequence of shocks at uncertain times and of uncertain size occurs.