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(Alvarez and Jermann (2000)) For each $i \in I$ there is a constant $\xi_i$ such that for all $t, z^t$, $$|u(c_{i,t}(z^t))| \le \xi_i u'(c_{i,t}(z^t)) c_{i,t}(z^t).$$ $\cdots$. This condition holds if (i) $u(\cdot)$ has relative risk aversion different from one at zero consumption, i.e.: $$\lim_{c \to 0} - \frac{c u''(c)}{u'(c)} \not=1,$$ which can be verified by repeated application of L'Hopital's rule, or (ii) $u'(0) < + \infty$, or (iii) if consumption for an agent is uniformly bounded away from zero.

It is Remark 1 in page 21 of working paper version, and page 784-5 in Econometrica. I don't understand how either (i), (ii), or (iii) imply the inequality condition. I was thinking maybe it has something to do with the concavity of $u(\cdot)$; i.e., $u(\xi) \le u(c_{i,t}(z^t)) + u'(c_{i,t}(z^t))(\xi - c_{i,t}(z^t))$. But, not sure about the next step. If you have any idea, can you give some help?

source: Working paper version, JSTOR.

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