Let's \(EU=X^\alpha\) be an individual's VNM expected utility function, where \(\alpha<1\), that is to say, the individual is risk averse. The individual's income is \(W_a\) with probability \(p\) and \(W_b\) with probability \((1-p)\). The individual has access to the insurance market, where he buys \(I\) units of insurance at the price \(s\) per dollar of insurance. The individual's expected utility maximization problem is
Max \(EU=p(W_a-sI)^{\alpha}+(1-p)[W_b+(1-s)I]^{\alpha}\) with respect to \(I\)
The first order condition yields
\(-\alpha s p (W_a-sI)^{\alpha-1}+\alpha (1-p)(1-s)[W_b+(1-s)I]^{\alpha-1}=0\)
Rearranging yields
\(sp(W_a-sI)^{\alpha-1}=(1-p)(1-s)[W_b+(1-s)I]^{\alpha-1}\)
Exponentiating both sides of the equation by \(\frac{1}{\alpha-1}\) yields
\((sp)^{1/\alpha-1}W_a-[(1-p)(1-s)]^{1/\alpha-1}W_b=[(sp)^{1/\alpha-1}+[(1-p)(1-s)]^{1/\alpha-1}]I\)
Solving for the optimal value of \(I^*\) gives us the individual's demand for insurance
\(I^*=\frac{(sp)^{1/\alpha-1}W_a-[(1-p)(1-s)]^{1/\alpha-1}W_b}{[(sp)^{1/\alpha-1}+[(1-p)(1-s)]^{1/\alpha-1}]}\)
Note that the demand for insurance is increasing in \(W_a\) and decreasing in \(W_b\), as well as increasing with respect to the probability of the event against which the individual is insuring (this is a bit roundabout to derive, but it follows from the fact that \(\alpha<1\).