Question

In: Economics

Suppose Alana has personal wealth of $10,000 and there is a probability of 0.2 of losing...

Suppose Alana has personal wealth of $10,000 and there is a probability of 0.2 of losing her car worth $6,400 in an accident. Her utility (of wealth) function is given by u(w) = w0.5, where w is wealth. Word limit per question: 400 words (200 words per part of question)

(a) What is Alana's expected wealth, expected utility, and utility of expected wealth? If she can insure "fully", and if this insurance is fair, how much would it cost her?

(b) What is the maximum amount Alana would be prepared to pay for full insurance? What is the certainty equivalent and the risk premium associated with the uncertain situation she is in if she does not have any insurance? What difference would it make if her utility of wealth function were instead u(w) = 5w?

Solutions

Expert Solution

Alana has personal wealth (w) of $10,000.

there is a probability of 0.2 of losing her car worth $6,400 in an accident. So, there is a probability of 0.2 that Alana's wealth becomes ($10,000-$6,400) = $3,600

And there is a probability of 0.8 that the accident does not happen and her wealth is $10,000

The utility function of Alana is given by:

A) Alana's expected wealth = 0.8*10,000 + 0.2*3600 = $8,000+$720 = $8,720

Alana's expected utility:

Utility of expected wealth = = 93.38

Fair price for full insurance = probability of loss * size of the loss = 0.2* $6,400 = $1,280

b) maximum amount Alana would be prepared to pay for full insurance = 10,000 - wealth associated with her expected utility (92) = 10,000 - (92)^2 =10,000-8,464 = $1,536

Calculate certainty equivalent by equating consumer's expected utility with uncertainty to that of consumer's expected utility without uncertainty. Let CE denote the certainty equivalent (what person get irrespective of the uncertainty).

consumer's expected utility with uncertainty = consumer's expected utility without uncertainty

CE =$ 8,464

Expected wealth  = $8,720

Risk premium = expected wealth - CE = 8720-8464= $256

If her utility function = u(w) = 5w, then

Alana's expected wealth = 0.8*10,000 + 0.2*3600 = $8,000+$720 = $8,720

then her expected utility :

Calculate certainty equivalent by equating consumer's expected utility with uncertainty to that of consumer's expected utility without uncertainty. Let CE denote the certainty equivalent (what person get irrespective of the uncertainty).

consumer's expected utility with uncertainty = consumer's expected utility without uncertainty

CE = 8,720

Risk premium = expected wealth - CE = 8720-8720= 0

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