4. 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: E[u(w+ e)] = u(w — T) where u is the agent's Bernoulli utility function, w is their wealth level, is the insurance premium/willingness to pay to avoid è, and è is mean-zero risk (i.e. è is a random variable with E[è] = 0). Prove that for small , r(w) = -u" (w)/u'(w) is proportional to . What is the constant of proportionality for this relationship? [Hint: start by taking the second- order Taylor expansion of the equation above].

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