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Let $c$ be the cost per unit of insurance, so the premium is equal to $cn$. Then the agent maximises: $$p(u(w - L - cn + n) + (1-p)u(w - cn).$$ The first order condition with respect to $n$ is given by: $$(1 - c) p u'(w - L - cn + n) - c (1-p) u'(w - cn) = 0$$ Rearranging gives:  \frac{p}{1 - p} = \frac{c}{1 - c} \frac{u'(w - cn)}{u'(w - L - cn + n)}. \$...