A farmer must decide whether to take protective action to limit damage to his grapefruit crop in the event that the overnight temperature falls to a level well below freezing. If the temperature drops too low he runs the risk of losing his entire crop, valued at $75,000. Based on the National Weather Service, the probability of such a temperature drop is 60%. He can insulate his crop by spraying water on all the trees, which will cost $20,000. This action might succeed in protecting the crop, with the following possible outcomes:

Probability Damage

0.30 $0 0.15 $5,000

0.10 $10,000

0.15 $15,000

0.30 $20,000

a) Construct a decision tree to help the farmer make his decision. What should he do? Explain your answer.

b) Find the highest cost of insulating the grapefruits for which the farmer prefers to insulate his crop.

c) Suppose the farmer is uncertain about the reliability of the National Weather Service forecast. If he thinks the probability of a freeze occurring could be anywhere between 40% and 80%, would that change his decision?

d) Now go back to original forecast of 60% chance of freeze. Suppose the farmer is not risk neutral, but instead his behavior can be modeled using an exponential utility function with a risk tolerance parameter of 100,000. What should he do, insulate or not? Explain your answer.

e) For the risk averse decision maker in part d, what is the certainty equivalent of his expected utility of insulating the crop?

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