Question

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 Refer to the scenario.

(A) Construct a decision tree to help the farmer make his decision.

(B) What should he do? Explain your answer.

(C) Refer to the scenario. Find the highest cost of insulating the grapefruits for which the farmer prefers to insulate his crop.

(D) Refer to the scenario. 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?

(E) Refer to the scenario. Suppose that, in addition to the uncertainty about the probability of freezing, he is also uncertain about the cost of the insulation (could vary from $10,000 to $30,000). To which of these variables is the expected value most sensitive?

(F) Refer to the scenario. 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 is the most he would be willing to pay for insulation in that case?

Answer 1

A. Two alternatives

Explanation:Alternative one : not to do anything, and run the risk of losing entire crop worth of $75,000.

Alternative two : to insulate his crop by spraying water at cost of $20,000 and succeed in saving the crop with five possible outcomes as given in attached file.

Net result or expected loss for atternative one is = .40*0 + .6*75000 = $45,000

Net result or expected loss for alternative two is = Cost-20000 + expected loss = 20000 + .3*0 +.15*5000 + .10*1000 +.15*15000 + .30*20000 = 20000+10000 = $30,000

B. Expected loss is less in case of alternative two, therefore farmer is advised to go for spraying water.

C. Highest cost for insulating the crop is (45000 - 10000) = $35,000 (Differnce between the expected loss under the two options)

D. In case the probability of freezing is at the lowest level of .40, then loss under alternative one is = .4*75000 =$30,000 . With higher levels of probabilities the expected loss is bound to be more. Therefore the decision to go for spraying water holds good.

E. As explained earlier as long as cost of insulation is less than the difference between the expected losses between options, it is recommended to go for spraying water.