Consider the stochastic incomplete market environment, assume endowments are iid and agents face a borrowing constraint where assets have to ge greater than $-\Phi$. Assume that $u$ is strictly increasing and concave.

The recursive problem of a household that has access to a risk-free bond that pays $R=1+r$ every period is \begin{align} v(a,s)=&\max_{c,a'}\left\{u(c)+\beta\sum_{s'}\pi(s')v(a',s')\right\}\\ s.t.\;\;& c+a'=(1+r)a+y_s\\ &a'\geq -\Phi \end{align}

How do I use the Dobb's convergence theorem to show that, using FOCs and envelope conditions, in general equilibrim, $\beta R<1$?



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