Can anyone explain the simple steps to solving any of these problems? Specifical

Question

Can anyone explain the simple steps to solving any of these problems?

Specifically "amount invested if they want to maximize their utility"

Suppose Feng have $50 and you would like to purchase some lottery tickets. Assume that he can win with 40% probability, and if he win, he can earn 100% of his investment (i.e., double the investment). On the other hand, if he lose, he loses 100% of his investment. Further assume that his utility function is given by u (w) = ?w2 + 100w. Answer each of the following.

(a) Is Feng risk-averse, risk-neutral, or risk-loving? How do you know?
(b) What is the expected value of the lottery if Feng invests all $50?
(c) If Feng want to maximize his expected utility, how much should he invest?

Now, suppose Han’s utility function is u(w)=ew?1.

(a) Is Han risk-averse, risk-neutral, or risk-loving? How do you know?
(b) What is the expected value of the lottery if Han invests all $50?
(c) If Han wants to maximize his expected utility, how much should he invest?

Now, suppose Harry’s utility function is u (w) = ew ? 1. He has a house of value $1,000,000. If earthquake comes, it will destroy his house and its value becomes zero. An insurance company provides insurance for those natural disasters. However, they cannot provide full insurance to the house. They provide insurance up to $500,000. The probability of the earthquake is 0.01%. The insurance fee is $0.02 per $1 insured asset. (i.e. if Harry insures $500,000, it will give him $500,000 if the earthquake happens. And the price for such insurance is $100 =$500, 000 × 0.02%.)

(a) Calculate the expected utility of Harry if he doesn’t purchase insurance.

(b) Would it be beneficial to Harry if he buy the insurance? If he dose, what is the optimal amount of insurance that he should buy?

(c) What is your answer to the question (b) if Harry’s utility function is u(w)=?w2+100w? Show your analysis.

Explanation / Answer

Ans a)

Risk Aversion Coefficinet is positive then Han is Risk averse

RA=-U"(w)U'(w)

U"(w)=-2w+100

U'(w)=100

RA=(100-2w)/100=1-0.02w

Hence agent is risk averse if w<50 and if w>50 then agent is risk liking

Ans b)

EMV=0.4(100)+0.6(0)=40

Ans c)

Lets differentiate utility function u'(w)=-2w+100

if -2w+100=0

w=50 now lets check whether utility would be maximized when w=50

after differentiating -2w+100 we get -2<0

hence w=50 utility will be maximised

Now u(w)=e^w-1

RA=-w(w-1)e^w-2/we^(w-1)

RA=-(w-1)/e

RA=(1-w)/e

When w<1 agent will be risk averse otherwise risk loving

b)

0.4(100)+0.6(0)=40

c)

as per part a) when w=1 then utility will be maximized

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