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 endowment. 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 complete market model 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 one experiences extinction relative to agent two, adding a third agent to the economy may reverse the situation and force the agent two to experience extinction relative to agent one. We also calculate the rates of convergence

A trace formula for functions of contractions and analytic operator Lipschitz functions

In this note we study the problem of evaluating the trace of $f(T)-f(R)$, where $T$ and $R$ are contractions on Hilbert space with trace class difference, i.e., $T-R\in\boldsymbol{S}_1$ and $f$ is a function analytic in the unit disk ${\Bbb D}$. It is well known that if $f$ is an operator Lipschitz function analytic in ${\Bbb D}$, then $f(T)-f(R)\in\boldsymbol{S}_1$. The main result of the note says that there exists a function $\boldsymbol{\xi}$ (a spectral shift function) on the unit circle ${\Bbb T}$ of class $L^1({\Bbb T})$ such that the following trace formula holds: $\operatorname{trace}(f(T)-f(R))=\int_{\Bbb T} f'(\zeta)\boldsymbol{\xi}(\zeta)\,d\zeta$, whenever $T$ and $R$ are contractions with $T-R\in\boldsymbol{S}_1$ and $f$ is an operator Lipschitz function analytic in ${\Bbb D}$.Comment: 6 page