# some derivation in Alvarez and Jermann (2000)

(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.