Relative risk aversion and lotteries exercise Given a consumer with a utility function, $u(w)$, and a wealth of $w&gt;1000$. Assuming that the consumers relative risk aversion is constant and equal to 1, that is $R_r(w)=1$ for $w&gt;0$, 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.
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 $u(w - p(w)) = U_L(w)$.