10. Bob must ship his belongings worth 1,000 to England from Latin America. The prob- ability that a ship sinks and he loses the entire value of his shipment is 0.1. Assume that if one ship sinks it does not affect the probability that the second ship sinks. He has the option of shipping everything in one single container or shipping his belong- ings separately on two different ships.
(a) [15 marks] If the shippers will transport Bob’s goods for the same price either separately or on the same ship, which method of shipment will he choose ac- cording to whether he is risk averse, risk loving or risk neutral? In your answer define and explain the concepts of risk aversion, risk loving and risk neutrality.
(b) [10 marks] Let Bob have utility U(X) = X for X 1,000 and U(X) = 500 + (X/2) for X > 1,000, where X is the value of the goods. Is Bob risk averse, risk neutral or risk loving? How much more would he be willing to pay for his preferred form of shipment?
(a) Risk aversion, risk loving, and risk neutrality are concepts used to describe an individual's attitude towards risk and uncertainty.
- Risk aversion: A risk-averse individual prefers certainty over uncertainty and is willing to pay a premium to avoid risky outcomes. They are not willing to take on higher risks even if the potential rewards are high.
- Risk loving: A risk-loving individual enjoys taking risks and is willing to endure the possibility of loss for the potential of higher gains. They are not concerned about potential losses and may even perceive a certain element of excitement or thrill in taking risks.
- Risk neutrality: A risk-neutral individual does not have a preference for risk-averse or risk-loving behavior. They evaluate options solely based on expected outcomes and do not consider the potential for gain or loss.
In this scenario, Bob faces the risk of losing the entire value of his shipment if a ship sinks. The probability of a ship sinking and the subsequent loss is 0.1.
If Bob is risk-averse, he would choose to ship his belongings separately on two different ships. By doing so, even if one ship sinks, he would still have the other shipment intact. This mitigates the risk of losing the entire value of his belongings.
If Bob is risk-loving, he may choose to ship everything in one single container. Despite the higher risk of losing the entire shipment, the potential reward of having both shipments arrive safely may outweigh the risk for him.
If Bob is risk-neutral, he would be indifferent between shipping his belongings separately or on the same ship. He would evaluate the options solely based on other criteria such as convenience or cost.
(b) Bob's utility function indicates that his level of satisfaction (utility) increases linearly with the value of his belongings, but at a decreasing rate. This utility function suggests that Bob is risk-averse.
To determine how much more Bob would be willing to pay for his preferred form of shipment, we need to compare the expected utility of each option.
1. Shipping everything in one single container:
- Expected utility = 0.9*U(1000) + 0.1*U(0) = 900 + 0 = 900
2. Shipping belongings separately on two different ships:
- Expected utility = 0.9*U(500) + 0.9*U(500) = 0.9*(500) + 0.9*(500) = 900 + 900 = 1800
Based on the expected utility calculations, Bob's preferred form of shipment is to ship his belongings separately on two different ships. He would be willing to pay a premium of 1800 - 900 = 900 for this option, as it provides twice the utility compared to shipping in one container.
(a) In order to determine which method of shipment Bob would choose, we need to understand the concepts of risk aversion, risk loving, and risk neutrality.
1. Risk Aversion: A risk-averse individual is someone who prefers certainty over uncertainty. They dislike taking risks and are willing to pay to reduce or avoid risks. In this case, if Bob is risk-averse, he would prefer to minimize the probability of losing the entire value of his shipment.
2. Risk Loving: A risk-loving individual is someone who enjoys taking risks and is willing to accept higher levels of uncertainty. They might even prefer taking risks with potentially higher rewards. If Bob is risk-loving, he might see the potential loss as an opportunity to gain more in case the shipment arrives safely. He might choose the riskier option in this case.
3. Risk Neutrality: A risk-neutral individual does not have a preference for taking or avoiding risks. They are indifferent to risk and base their decisions solely on expected values. If Bob is risk-neutral, he would be indifferent between the two methods of shipment as long as the expected values are the same.
Now, let's consider the options for Bob and determine which method he would choose according to his risk preference:
- Shipping everything in one single container: The probability of losing the entire value of his shipment is 0.1. Therefore, the expected value of his shipment is (1 - 0.1) * 1,000 = 900.
- Shipping belongings separately on two different ships: Since the probability of one ship sinking does not affect the probability of the other, the probability of losing the entire value of his shipment would be (0.1) * (0.1) = 0.01. Therefore, the expected value of his shipment is (1 - 0.01) * 1,000 = 990.
If the shippers charge the same price for both methods, Bob, being risk-averse, would prefer shipping everything in one single container. This is because the expected value of his shipment is higher (900) compared to shipping separately (990).
(b) To determine whether Bob is risk-averse, risk-neutral, or risk-loving, we need to analyze his utility function and his preferences towards risk.
- Bob's utility function is defined as U(X) = X for X ≤ 1,000 and U(X) = 500 + (X/2) for X > 1,000. This means that Bob has a linear utility function for values less than or equal to 1,000 and an increasing but diminishing utility function for values greater than 1,000.
- If Bob is risk-averse, he would be willing to pay more to avoid the risk. In this case, since he prefers shipping everything in one single container (with an expected value of 900), we can calculate how much more he would be willing to pay compared to the riskier option. Let's assume the price for either method is $P. Bob would be willing to pay (990 - 900) = $90 more for his preferred form of shipment.
Therefore, based on Bob's utility function and his preference for shipping everything in one single container, we can conclude that he is risk-averse and is willing to pay an additional $90 for his preferred form of shipment.