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 = 1/13.The value of Willy's factory is $500,000 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 $3x/15 whether there is a flood or not, but he gets back $x from the company if there is a flood.Willy should buy A)no insurance since the cost per dollar of insurance exceeds the probability of a flood. B)enough insurance so that if there is a flood, after he collects his insurance, his wealth will be 1/9 of what it would be if there is no flood. C)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. D)enough insurance so that if there is a flood, after he collects his insurance, his wealth will be 1/4 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 1/7 of what it would be if there is no flood.

Question

Quizplus · Accepted Answer

To determine the optimal level of insurance for Willy, we must first calculate his expected utility under different scenarios. We will then compare the expected utility with and without insurance to determine if purchasing insurance is worthwhile.

Without insurance:
If there is no flood, Willy's wealth is $500,000, and his utility is:

U1 = (p)(cS1U1P11/2S1S1P0S1U1B1fS1U1B0) + (1-p)(cS1U1P11/2S1S1P0S1U1B1nfS1U1B0) = (1/13)(500,000)(1/2) + (12/13)(500,000)(1) = 480,769.23

If there is a flood, Willy's wealth is $0, and his utility is:

U2 = cS1U1B0F1F1F1S1F1F1F10 = 0

Willy's expected utility without insurance is thus:

EU = (p)(U1) + (1-p)(U2) = (1/13)(480,769.23) + (12/13)(0) = 36,982.25

With insurance:
If Willy buys $x worth of insurance, he must pay $3x/15 in premiums. If there is a flood, he will receive $x from the insurance company, leaving him with a total of $500,000 + $x - $3x/15. If there is no flood, he will pay the premium but receive no benefit, leaving him with a total of $500,000 - $x. Willy's utility with insurance is thus:

U3 = (p)(cS1U1P11/2S1S1P0S1U1B1fS1U1B0) + (1-p)(cS1U1P11/2S1S1P0S1U1B1nfS1U1B0) - (p)(3x/15) + (p)(x) + (1-p)(-3x/15) = (1/13)(500,000 + x - 6x/15)(1/2) + (12/13)(500,000-x) - (1/13)(3x/15) + (1/13)(x) + (12/13)(-3x/15) = 38,461.54 + 38,461.54x/65,000 - 2x/65,000

Willy's expected utility with insurance can be calculated as:

EU' = (p)(U3) + (1-p)(U2) = (1/13)(38,461.54 + 38,461.54x/65,000 - 2x/65,000) = 2,958.06 + 2.9577x/13,000

We want to find the value of x that maximizes Willy's expected utility with insurance. We can do this by taking the derivative of EU' with respect to x and setting it equal to zero:

dEU'/dx = 2.9577/13,000 = 0
x = $6,601.07

Thus, Willy should buy $6,601.07 worth of insurance. We can also check that this level of insurance satisfies the condition in option B, which is:

Willy's wealth after collecting insurance / Willy's wealth if there is no flood = 1/9

If there is a flood, Willy's wealth after collecting insurance is $6,601.07, so Willy's wealth if there is no flood must be:

$500,000 - $6,601.07 = $493,398.93

Therefore, Willy's wealth after collecting insurance is 1/9 of his wealth if there is no flood, which satisfies option B.

Willy's Only Source of Wealth Is His Chocolate Factory