Willy's only source of wealth is his chocolate factory.He has the utility function pc^1/2_f + (1 - p)c^1/2_nf, where p is the probability of a flood, 1 - p is the probability of no flood, and c_f and c_nf are his wealth contingent on a flood and on no flood, respectively.The probability of a flood is p = 1/14.The value of Willy's factory is $400,000 if there is no flood and 0 if there is a flood.Willy can buy insurance where if he buys $x worth of insurance, he must pay the insurance company $5x/18 whether there is a flood or not, but he gets back $x from the company if there is a flood.Willy should buy A)enough insurance so that if there is a flood, after he collects his insurance, his wealth will be the same whether there is a flood or not. B)enough insurance so that if there is a flood, after he collects his insurance, his wealth will be 1/6 of what it would be if there is no flood. C)no insurance since the cost per dollar of insurance exceeds the probability of a flood. D)enough insurance so that if there is a flood, after he collects his insurance, his wealth will be 1/25 of what it would be if there is no flood. E)enough insurance so that if there is a flood, after he collects his insurance, his wealth will be 1/11 of what it would be if there is no flood.

Question

Quizplus · Accepted Answer

Willy's expected utility without insurance is:
$$EU=pcS_1U_1P_1\frac{1}{2}S_1S_1P_0S_1U_1B_1fS_1U_1B_0+F_1F_1F_1S_1+F_1F_1F_10$$
Substituting in numerical values, we get:
$$EU=\frac{1}{14}\cdot400,000\cdot\sqrt{cS_1U_1B_1fS_1U_1B_0}+\frac{13}{14}\cdot400,000\cdot\sqrt{cS_1U_1B_1nfS_1U_1B_0}$$
Taking partial derivatives of the expected utility with respect to $cS_1U_1B_1fS_1U_1B_0$ and $cS_1U_1B_1nfS_1U_1B_0$, and setting them equal to zero, we get:
$$\frac{\partial EU}{\partial cS_1U_1B_1fS_1U_1B_0}=\frac{\partial EU}{\partial cS_1U_1B_1nfS_1U_1B_0}$$
$$\frac{p\cdot200,000}{14\cdot\sqrt{cS_1U_1B_1fS_1U_1B_0}}=\frac{(1-p)\cdot200,000}{14\cdot\sqrt{cS_1U_1B_1nfS_1U_1B_0}}$$
Simplifying, we get:
$$\sqrt{\frac{cS_1U_1B_1fS_1U_1B_0}{cS_1U_1B_1nfS_1U_1B_0}}=\sqrt{\frac{1-p}{p}}$$
$$\frac{cS_1U_1B_1fS_1U_1B_0}{cS_1U_1B_1nfS_1U_1B_0}=\frac{1-p}{p}$$
$$cS_1U_1B_1fS_1U_1B_0=\frac{(1-p)}{p}cS_1U_1B_1nfS_1U_1B_0$$
Willy's expected utility with insurance where he buys $x$ worth of insurance is:
$$EU(p,x)=(1-p)\cdot\left[400,000-\frac{5x}{18}+x\right]+\frac{p}{14}\cdot\left[400,000-\frac{5x}{18}\right]$$
Simplifying, we get:
$$EU(p,x)=(1-p)\cdot\left(400,000+\frac{13x}{18}\right)+\frac{p}{14}\cdot\left(400,000-\frac{5x}{18}\right)$$
Taking the partial derivative of expected utility with respect to $x$ and setting it equal to 0, we get:
$$\frac{\partial EU(p,x)}{\partial x}=-\frac{13(1-p)}{18}+\frac{5p}{252}=0$$
Solving for $p$, we get $p=\frac{13}{25}$.
Substituting this value of $p$ back into the expected utility equation, we get:
$$EU(x)=\frac{12}{25}\cdot(400,000+\frac{13x}{18})+\frac{13}{350}\cdot(400,000-\frac{5x}{18})$$
Simplifying, we get:
$$EU(x)=\frac{52}{25}\cdot10^4+\frac{17}{210}\cdot x$$
Willy wants to buy enough insurance such that if there is a flood, after he collects his insurance, his wealth will be 1/25 of what it would be if there is no flood. This means that his wealth after a flood should be:
$$\frac{1}{25}\cdot400,000=\frac{1}{25}\cdot\left(400,000-\frac{5x}{18}+x\right)$$
Solving for $x$, we get $x=\frac{8}{3}\cdot10^4$.
Therefore, Willy should buy enough insurance so that he pays $\frac{5}{18}\cdot\frac{8}{3}\cdot10^4=\frac{4}{3}\cdot10^4$ to the insurance company, and if there is a flood, he collects $\frac{8}{3}\cdot10^4$ from the company. This way, his expected utility is maximized.

Willy's Only Source of Wealth Is His Chocolate Factory