Relative Extinction of Heterogeneous Agents

In all the existing literature on survival in heterogeneous economies, the rate at which an agent vanishes in the long run relative to another agent can be characterized by the difference of the so-called survival indices, where each survival index only depends on the preferences of the corresponding agent and the properties of the aggregate en- dowment. In particular, one agent experiences extinction relative to another (that is, the wealth ratio of the two agents goes to zero) if and only if she has a smaller survival index. We consider a simple continuous-time model of the Merton-Black-Scholes type and show that the survival index is more complex if there are more than two agents in the economy. In fact, the following phenomenon may take place: even if agent i experiences extinction relative to agent j, adding a third agent k to the economy may reverse the situation and force the agent j to experience extinction relative to agent i. We also calculate the rate of convergence.