1. **Problem statement:** We analyze expected loss and insurance choices for Henry and Lucy given initial wealth $W_0=4975$ and a loss of 85% of wealth with probability 0.1. 2. **Expected loss (no insurance):** $$\text{Loss} = 0.85 \times 4975 = 4238.75$$ Expected loss: $$E[\text{loss}] = 0.1 \times 4238.75 = 423.875$$ 3. **Henry's certainty-equivalent loss:** Henry's utility is $u_H(x) = \sqrt{x}$. Expected utility without insurance: $$EU = 0.9 \times \sqrt{4975} + 0.1 \times \sqrt{736.25}$$ Calculate: $$\sqrt{4975} \approx 70.534, \quad \sqrt{736.25} \approx 27.141$$ So: $$EU \approx 0.9 \times 70.534 + 0.1 \times 27.141 = 66.191$$ Certainty-equivalent wealth $CE$ satisfies: $$\sqrt{CE} = 66.191 \implies CE = (66.191)^2 \approx 4381.240$$ Sure loss equivalent: $$4975 - 4381.240 = 593.760$$ 4. **Lucy’s optimal insurance choice ($p=0.1$):** Lucy’s utility is linear: $u_L(x) = x$. Expected wealth gain per unit insurance: $$0.1 \times 1 - 0.1 = 0$$ Lucy is indifferent across all insurance levels $\alpha$. The largest such choice is $\infty$. 5. **Lucy’s optimal insurance choice ($p=0.2$):** Expected gain per unit: $$0.1 \times 1 - 0.2 = -0.1$$ Negative expected value means buy none: $$0.000$$ 6. **Henry’s optimal insurance choice ($p=0.2$):** Final wealth: No loss: $4975 - 0.2\alpha$ Loss: $736.25 + 0.8\alpha$ Expected utility: $$EU = 0.9 \sqrt{4975 - 0.2\alpha} + 0.1 \sqrt{736.25 + 0.8\alpha}$$ First order condition (FOC) yields: $$\alpha \approx 293.382$$ **Final answers:** - Expected loss (no insurance): $423.875$ - Henry’s certainty-equivalent loss: $593.760$ - Lucy’s optimal insurance ($p=0.1$): $\infty$ - Lucy’s optimal insurance ($p=0.2$): $0.000$ - Henry’s optimal insurance ($p=0.2$): $293.382$