28536 Choice Under Uncertainty Case Study - Economics Assignment Help

Assignment Task

Task

Uncertainty:

In this sub-topic we are focusing on consumer choice under uncertainty. We begin this sub-topic by introducing and analysing two key concepts: expected value or expected gain and expected utility.Knowing and comparing consumer’s utility of the expected value and the expected utility enables us to differentiate consumer’s attitude to risk. Having all these key elements and applying a reasoning symmetrical to consumer’s choice in certainty (topic 1) leads us to finding consumer demand under uncertainty. The typical examples of analysis include insurance market.One of the key “tricks” to solve problem sets on consumer decisions under uncertainty is to treat consumption possibilities – consumption plans – at different dates as different commodities. The next important step is to be able to find state-contingent budget constraint.

Topic 1 Discussion questions.

Q1. Discuss the statement “If the price of insurance goes up, people will become less risk averse.” Is this statement true or false?

Q2. If someone has strictly convex preferences between all contingent commodity bundles, then is this consumer risk-loving, risk-averse or risk-neutral?

Q3. Suppose a standard utility maximiser’s preferences between two bundles so that the first one is contingent on event 1 happening and second one contingent on event 2 happening are independent.What can we say about these preferences and how can we describe them.

Q4. A consumer has a von Neumann-Morgenstern utility function of the form U(cA,cEopA,p8) = pAv(cA) + pav(c8),, where pA and pB are the probabilities of events A and B and where cA and cB are consumptions contingent on events A and B respectively. Must this consumer be a risk lover if v is an increasing function?

Q5. Harley’s current wealth is 600, but there is a 0.25 probability that he will lose 100. Harley is risk neutral. He has an opportunity to buy insurance that would restore his 100 if he lost it.

(a) Harley would be willing to pay a bit more than 25 for this insurance.

(b) Harley would be willing to pay up to 25 for this insurance.

(c) Since Harley is risk neutral, he wouldn’t be willing to pay anything for this insurance.

(d) Since Harley’s utility function is not specified, we can’t tell how much he would be willing to pay for this insurance.

(e) Harley would not be willing to pay more than 16.66 for this insurance.

Q6. Mabel and Emil were contemplating marriage. They got to talking. Mabel said that she always acted according to the expected utility hypothesis, where she tried to maximise the expected value of the log of her income. Emil said that he too was an expected utility maximiser, but he tried to maximise the expected value of the square of his income. Mabel said, “I fear we must part. Our attitudes toward risk are too different.” Emil said, “Never fear, my dear, the square of income is a monotonic increasing function of the log of income, so we really have the same preferences.” Who is right about whether 
their preferences toward risk are different?

(a) Mabel is right.

(b) Emil is right.

(c) Emil is right about small risks but wrong about large risks.

(d) Mabel is right about small risks but wrong about large risks.

(e) They are both wrong

Topic 2 Problem sets.

1. Calculate the expected utility of a football player F, if her von Neumann-Morgenstern utility function is is U(c) = c112; her income if she is injured is 10,000 and her income is she is not injured is 25 million. Her probability of being injured is 0.1.

2. Portia has waited a long time for her ship to come in, and she has concluded that it will arrive today with probability 1/4. If it does come, she will receive 16. If it doesn’t come in today, it never will and she will have zero wealth. She has a von Neumann-Morgenstern utility function equal to the square root of her total income. What is the minimum price at which she would sell the rights to her ship?

3. Joe’s wealth is 100 and he is an expected utility maximiser with a von Neumann-Morgenstern utility function function U(W) = W1/12. Joe is afraid of oversleeping his economics exam. He figures there is only a 1 in 10 chance that he will, but if he does, it will cost him 100 in fees to the university for taking an exam late. Joe’s neighbour, Mary, never oversleeps. She offers to wake him one hour before the test, but he must pay her for this service. What is the most that Joe would be willing to pay for this wake-up service?

4. The certainty equivalent of a gamble is defined to be the amount of money which, if you were promised it with certainty, would be indifferent to the gamble.

(a) If an expected utility maximiser has a von Neuman-Morgenstern utility function U(W) = W1/12.(where W is wealth) and if the probability of events 1 and 2 are both ½, write a formula for the certainty equivalent of a gamble that gives you x if event 1 happens and y if event 2 happens.

(b) Generalise your formula in part (a) to the case where the probability of event 1 is p and the probability of event 2 is 1 - p.

(c) Generalise the formula in part (a) to the case where U(W)=Wa for >0

5. Suppose the owner of a factory, David. This factory is located close to a river and may suffer from floods in the spring. Suppose now the owner of the factory wants to sell it and retire in the summer.If there is no spring flooding, the factory will be worth 500,000. Otherwise, the destroyed factory will be worth only 50,000. David can buy flood insurance that costs 0.1 for each 1 of wealth ensured. David estimates the probability of the flood in the next spring to be 10%. He is also rational maximiser of expected utility with von Neumann-Morgenstern utility function 0. 1cF12 + 0.9cNF12, where cF stands for contingency consumption plan in case of a flood and cNF for consumption plan if there was no flood this spring. Find David’s optimal consumption bundle and the amount of insurance premium.

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