# How a utility function which is both DARA and CRRA can be explained?

I'm studying risk aversion and I cannot make a intuitive explain about the utility function which is DARA and CRRA.

for instance, let's say, $\ln W$, where $W$ stands for one's wealth.

by the definition of absolute risk aversion, its ABA is ${1}/{W}$ and it is CRRA since its relative risk aversion is $1$ .

then here's the start of my confusion.

first, consider a fair gambling, and two guys who have same utility function as $\ln W$. By DARA, the one who is richer would not feel much riskier than the poor. So he would not spend as much as the poor guy do for making fix his income, for example by contracting insurance(it means the richer would pay less insurance fee for hedging his risk than the poor guy.

second, what about the aspect of RRA(relative risk aversion)? It is constant. I understood this as,, that the rich man and the poor man's spending in their insurance fee does not depend on their wealth(specifically, does not vary when their wealth increases or decreases) let's say for some risk they would like to pay 3% of their wealth to avoid the certain risk. Then, from here, it is clear that the 3% of wealth of rich man is bigger than the poor man.

In other words, It looks that the rich man would pay much more money for insurance than the poor man when we see the aspect of RRA. However, DARA says that it is not the case. For me, this does not makes sense. And with a common sense, if a person who is willing to avoid risk would feel less risky when he becomes rich. So DARA looks fine. But the situation with CRRA looks contradiction about one's utility (function).

The question should have been shorter..sorry about that. Please leave me a hint to solve this confusion. Thanks.

• "Risk aversion", "absolute", "relative" are just labels, not necessarily illuminating, and maybe even confusing. Do you know the rationale behind these measures? How they have been originally conceived, rationalized and obtained? – Alecos Papadopoulos Apr 14 '17 at 15:03
• I would also suggest to read the exposition regarding the Absolute Risk Aversion measure in Jehle's and Reny Advanced Microeconomic Theory book. – Alecos Papadopoulos Apr 14 '17 at 15:22