# Using risk aversion

I'm trying to figure out what the non-stochastic equivalent payment is for someone who is risk-averse. Suppose we have a lottery that pays out\$100 with probability one half and \$0 with probability one half. If we are using a coefficient of relative risk aversion of 2, then we have a utility function of the form $$u(w) =\frac {w ^ {1-2} -1} {1-2} = -w ^ {-1} +1$$

This seems very weird to me because utility is negatively correlated with wealth. What am I doing wrong?

(I am specifically asking because I want to understand this paper, which uses a similar utility function on page 18)

• My two cents (beyond the fact that in the page you linked, the isoelastic utility function you use is introduced in the context of intertemporal decision-making): you pick $R(w) = 2$, but you cannot really impose it, right? This should be because $R(w) = w A(w) =$ etc etc, as written in the wiki entry. Sep 1, 2015 at 14:02
• Thanks – I added a link to the paper I am actually trying to understand, which does have hardcoded values for this coefficient (I believe). Let me know if I misunderstanding this as well! Sep 1, 2015 at 14:20
• Ok, I took a look. Well, I think what I wrote about imposing the value $R(w) =2$ still stands. To me, you cannot really do it, because that value does depend on $w$ as well, as you can see in the wiki entry. Sep 1, 2015 at 15:20
• It seems to me that in the paper cited in table 5 they have a hardcoded value (which they denote $\gamma$) which they plug into equation (10). Are you saying that I am misunderstanding what they did, or that what they did is wrong? Sep 1, 2015 at 16:02
• Of course I did not write the paper, but I simply went to the page you pointed out to me and I try to roughly see the context. Hence, I am not saying they did anything wrong. What I am saying is that I think that their $\gamma$ is defined as the $\rho$ in the wiki entry you linked. Hence, it is in function of something. Sep 1, 2015 at 18:40

What do you mean with "utility is negatively correlated with wealth"? The utility function $$1- 1/w$$ is increasing in w. See http://www.wolframalpha.com/input/?i=Plot[1-1%2Fx%2C{x%2C0%2C100}]
However, it is not defined at $w=0$. So you need some initial wealth You can calculate $$\frac{1}{2} u(w+0) + \frac{1}{2} u(w+100) = u (w+x)$$ Then, $x$ is the deterministic payment such that the agent is indifferent between the lottery and the safe payment of $x$.