Question

Aaron is a farmer and faces the following uncertain situation. He has a 20% chance of having a poor harvest where he will lose $ 5000 and an 80% chance of having a good harvest where he will gain $ 10,000. Aaron has a current wealth of $ 80,000 and a utility function of the form U(W) = ln W.

1. What is the expected value of his final wealth in this situation?
2. What is the expected utility of this uncertain situation? Draw the appropriate graph to demonstrate the situations in parts (A) and (B) of this problem.
3. What is the certainty equivalent for this uncertain situation? Show this on your graph.
4. What is the risk premium that Aaron would pay to avoid this uncertain situation? Show this on your graph.

Answer 1

A.

Income in poor state = 80000 - 5000 = 75,000

Income in good state = 80,000 + 10000 = 90,000

Expected Value = Probability in good state*Gain in good state + probability in bad state * Loss in bad state

= 0.2*75,000 + 0.8*90,000

= 15000 + 72000

= 87000

B.

Expected Utility, EU = 0.2U(75000) + 0.8*(90000)

= 0.2*ln(75000) + 0.8*ln(90000)

= 0.2*11.23 + 0.8*11.41

= 2.246 + 9.128

= 11.37

C.

For certainty equivalent, CE, we have

U(CE) = EU

ln (CE) = 11.37

CE = antilog(11.37)

CE = 86,681.87

D.

Risk Premium = EV - CE = 87000 - 86,681.87

RP = 318.13