Question

Suppose a person has the utility function, U(I)=log(I), where I is income. He has income I2 ($4,000) with the probability of p, but knows that some externally generated risk may reduce his income to I1 ($1,000) with probability of 1-p. Suppose p=0.8.

1) Is this person risk-averse? If so, why?

2) What is the expected income of this person?

3) What is the utility of expected income for this person?

4) What is the expected utility of this person?

5) Compare 3) and 4). Which is larger? Why?

6) What is the certainty equivalent income in this case?

7) What is the risk premium in this case?

8) Graphically show the risk premium with this utility function. Make sure to label each number of I1 and I2, expected income, expected utility, certainty equivalent income, together with the risk premium on the same graph.

9) Calculate the risk premium if p=0.5.

10) Is risk premium in 9) smaller or larger than the one you get in 7)? Why? Briefly summarize the intuition behind this result.

Answer 1

As per the Chegg Policy, answering the first questions

U(I) = log(I)

1)

A person is risk-averse as the utility function is concave.

2)

Expected Income = 0.8*4000 + 0.2*1000

= 3200 + 200

= $3400

3)

Utility of Expected Income: U(3400) = log(3400) = 3.53

4)

in good state, income = $4000
In bad state, loss of $3000
So income in bad state = 4000 - 3000 = $1000
Bad state probability = 1 - p = 0.2, good state, p = 0.8
Thus EU = .8*ln(4000) + .2*ln(1000)

EU = .8 * 3.6 + .2 * 3

EU = 2.88 + 0.6

EU = 3.48

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