1. **Problem statement:** Lucy and Henry each have 4975 units of wealth. With probability 0.1, they lose 85% of their wealth. They can buy $\alpha$ units of insurance at cost 0.1 per unit, each paying 1 unit if loss occurs. Lucy's utility is $u_L(x) = x$, Henry's utility is $u_H(x) = \sqrt{x}$. We answer parts a) to e). 2. **Part a) Expected value of the loss without insurance:** The loss amount is 85% of 4975: $$\text{Loss} = 0.85 \times 4975 = 4228.75$$ Expected loss: $$E[\text{Loss}] = 0.1 \times 4228.75 + 0.9 \times 0 = 422.875$$ 3. **Part b) Henry's certainty equivalent loss:** Henry's expected utility without insurance is: $$E[u_H] = 0.1 \times \sqrt{4975 - 4228.75} + 0.9 \times \sqrt{4975} = 0.1 \times \sqrt{746.25} + 0.9 \times \sqrt{4975}$$ Calculate: $$\sqrt{746.25} \approx 27.32, \quad \sqrt{4975} \approx 70.52$$ So: $$E[u_H] = 0.1 \times 27.32 + 0.9 \times 70.52 = 2.732 + 63.468 = 66.2$$ Henry's utility if he loses $x$ for sure is: $$\sqrt{4975 - x} = 66.2$$ Square both sides: $$4975 - x = 66.2^2 = 4381.44$$ Solve for $x$: $$x = 4975 - 4381.44 = 593.56$$ 4. **Part c) Lucy's utility maximizing $\alpha$ with insurance cost 0.1:** Lucy’s wealth if loss occurs and she buys $\alpha$ units: $$W_{loss} = 4975 - 0.85 \times 4975 + \alpha - 0.1 \alpha = 4975 - 4228.75 + 0.9 \alpha = 746.25 + 0.9 \alpha$$ If no loss: $$W_{no\ loss} = 4975 - 0.1 \alpha$$ Expected utility: $$EU_L = 0.1 (746.25 + 0.9 \alpha) + 0.9 (4975 - 0.1 \alpha) = 74.625 + 0.09 \alpha + 4477.5 - 0.09 \alpha = 4552.125$$ Notice $\alpha$ terms cancel out, so expected utility is constant regardless of $\alpha$. Since utility is linear, Lucy is indifferent to $\alpha$ and the largest $\alpha$ is unbounded. But practically, $\alpha$ cannot exceed the loss amount 4228.75. So the largest $\alpha$ maximizing utility is: $$\alpha = 4228.75$$ 5. **Part d) Lucy's utility maximizing $\alpha$ with insurance cost 0.2:** Now cost per unit is 0.2, so: $$W_{loss} = 746.25 + \alpha - 0.2 \alpha = 746.25 + 0.8 \alpha$$ $$W_{no\ loss} = 4975 - 0.2 \alpha$$ Expected utility: $$EU_L = 0.1 (746.25 + 0.8 \alpha) + 0.9 (4975 - 0.2 \alpha) = 74.625 + 0.08 \alpha + 4477.5 - 0.18 \alpha = 4552.125 - 0.1 \alpha$$ Since $EU_L$ decreases with $\alpha$, Lucy maximizes utility by choosing the smallest $\alpha$, which is 0. 6. **Part e) Henry's utility maximizing $\alpha$ with insurance cost 0.2:** Henry's expected utility: $$EU_H = 0.1 \sqrt{746.25 + 0.8 \alpha} + 0.9 \sqrt{4975 - 0.2 \alpha}$$ We find $\alpha$ maximizing $EU_H$ on $[0, 4228.75]$. Derivative: $$\frac{dEU_H}{d\alpha} = 0.1 \times \frac{0.8}{2 \sqrt{746.25 + 0.8 \alpha}} - 0.9 \times \frac{0.2}{2 \sqrt{4975 - 0.2 \alpha}} = \frac{0.04}{\sqrt{746.25 + 0.8 \alpha}} - \frac{0.09}{\sqrt{4975 - 0.2 \alpha}}$$ Set derivative to zero: $$\frac{0.04}{\sqrt{746.25 + 0.8 \alpha}} = \frac{0.09}{\sqrt{4975 - 0.2 \alpha}}$$ Square both sides: $$\frac{0.0016}{746.25 + 0.8 \alpha} = \frac{0.0081}{4975 - 0.2 \alpha}$$ Cross multiply: $$0.0016 (4975 - 0.2 \alpha) = 0.0081 (746.25 + 0.8 \alpha)$$ Calculate: $$7.96 - 0.00032 \alpha = 6.045 + 0.00648 \alpha$$ Bring terms together: $$7.96 - 6.045 = 0.00648 \alpha + 0.00032 \alpha$$ $$1.915 = 0.0068 \alpha$$ Solve for $\alpha$: $$\alpha = \frac{1.915}{0.0068} \approx 281.62$$ Check endpoints: $$EU_H(0) = 0.1 \times 27.32 + 0.9 \times 70.52 = 66.2$$ $$EU_H(4228.75) = 0.1 \times \sqrt{746.25 + 0.8 \times 4228.75} + 0.9 \times \sqrt{4975 - 0.2 \times 4228.75} = 0.1 \times \sqrt{746.25 + 3383} + 0.9 \times \sqrt{4975 - 845.75} = 0.1 \times \sqrt{4129.25} + 0.9 \times \sqrt{4129.25} = \sqrt{4129.25} \approx 64.26$$ Since 66.2 > 64.26 and the critical point is a maximum, the maximizing $\alpha$ is approximately 281.62. **Final answers:** - a) Expected loss = 422.875 - b) Henry's certainty equivalent loss = 593.56 - c) Lucy's optimal $\alpha$ at cost 0.1 = 4228.75 - d) Lucy's optimal $\alpha$ at cost 0.2 = 0 - e) Henry's optimal $\alpha$ at cost 0.2 = 281.62 All answers rounded to three decimals where applicable.