Question

6. Consider an individual whose utility function over money is u(w)= 1+2w2. (a) Is the individual risk-averse, risk-neutral, or risk-loving? Does it depend on w? (b) Suppose the individual has initial wealth ¥W and faces the possible loss of Y". The probability that the loss will occur is . Suppose insurance is available at price p, where p is not necessarily the fair price. Find the optimal amount of insurance the individual should buy. You may assume that the solution is interior. (c) Is there a price at which the individual will not want to buy any insurance? If so, find it. If no, explain.

Answer 1

Ans (a):

Based on utility function over money, we can compute the utility curve and based on its shape we can know the risk appetite of the individual.

U(W)

A risk averse utility function is concave (red shaded curve above), risk neutral is upward sloping without curvature (grey shaded curve above), and a risk loving utility function is convex (orange shaded curve above).

Based on the given utility function, we can plot the utility curve as follows:

Utility function over money Utility Value 1 10 20 30 40 50 60 Wealth/Money (W)

Therefore, the individual is risk neutral, since the curve is not concave or convex.

Ans (b):

Given:

Initial wealth (W_O)= W

Probable Loss (L) = W/2

Probability of Loss (P_L) = ½

Probability of No Loss (P_N) = ½

Insurance Premium = p

Applying the following formula, we can know the amount of Insurance coverage (C), assuming loss occurs:

L = W_O – (P_L * C * L) – L + (C * L)

W/2 = W – (1/2 * C * W/2) – W/2 + (C * W/2)

After solving for C, we find C = 0

Therefore, no insurance is needed, in this scenario

Ans (c):

As per the last question, we understood that there is’nt a need for any insurance cover in this case. Therefore, an individual would not want to pay any premium for it. Therefore, an individual would only want to buy an insurance at zero premium.