(Solved): (Extra credit) In class we talked about how the Arrow-Pratt coefficient of ab ...

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(Extra credit) In class we talked about how the Arrow-Pratt coefficient of absolute risk aversion can be thought of as proportional to the insurance premium that an expected utility maximizer would be willing to pay to completely avoid a small, mean zero risk. Mathematically, we could write this insight the following way: $\mathbb{E}[u(w+\widetilde{?})]=u(w??)$ where $u$ is the agent's Bernoulli utility function, $w$ is their wealth level, $?$ is the insurance premium/willingness to pay to avoid $\widetilde{?}$, and $\widetilde{?}$ is mean-zero risk (i.e. $\widetilde{?}$ is a random variable with $\mathbb{E}[\widetilde{?}]=0)$. Prove that for small $\widetilde{?},r(w)=?u^{??}(w)/u^{?}(w)$ is proportional to $?$. What is the constant of proportionality for this relationship? [Hint: start by taking the secondorder Taylor expansion of the equation above].