# Relative risk aversion and lotteries exercise

Given a consumer with a utility function, $u(w)$, and a wealth of $w>1000$. Assuming that the consumers relative risk aversion is constant and equal to 1, that is $R_r(w)=1$ for $w>0$, the consumer is facing a lottery that gives him 50% chance to win 1000 and 50% chance to lose 1000.

My question is, how much is the consumer willing to pay to avoid this lottery, and how does it depend on $w$? I can't seem to figure it out, and I really have no idea where to start.

• Start here. Than graph your u(w). Make sure that w>1000 is a correct condition. It seems strange. May be, u(w) is defined on w>0 and his/her current wealth is a point in a range w>1000? May 31, 2015 at 13:21
• $u(w)$ is defined for $w>0$ - his current wealth is $w>1000$ when he is facing the lottery. Sorry I didn't make that clear :) I'll have a look at your link, thanks. May 31, 2015 at 13:23
• You should edit your question, and my comment will be redundant. And do not forget to carefully read site rules =)) May 31, 2015 at 13:26

You want the household to be ex-ante indifferent between taking the lottery and paying some $p$ and voiding it.
With van-Neumann-Morgenstein utility, the utility of taking the lottery is given by $$U_L(w) \equiv 0.5 U(w-1000) + 0.5U(w+1000)$$
Now, you're looking for a payment $p(w)$ such that $u(w - p(w)) = U_L(w)$.