ECO6007F – Microeconomics II – Tutorial I (Risk)

Question 1 A woman with current wealth X has the opportunity to bet any amount on the occurrence of an event that she knows will occur with probability p. If she wagers w, she will receive 2w if the event occurs and 0 if it does not. Assume that her Bernoulli utility function takes the following form:

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u(x) = −e−rx , with r > 0. 1.1 How much should she wager? (Hint: the amount depends on p) 1.2 Does her utility function exhibit CARA, DARA or IARA? Question 2 Alex plays football for a local club in Cape Town. If he does not suffer any injury by the end of the season, he will get a professional contract with Kaiser Chiefs, which is worth R10 000. If he is injured though, he will get a contract as a fitness trainer in Cape Town which is worth only R100. The probability of suffering an injury is 10%. 2.1 Describe this lottery 2.2 What is the expected value of this lottery? 2.3 What is the expected utility of this lottery if u(x) = x 2.4 Assume he could buy insurance at price ρ that would pay him R9 900 in the case of an injury. What is the highest value of ρ that makes it worthwhile for Alex to purchase insurance? 2.5 What is the certainty equivalent for this lottery? 2.6 Using the Bernoulli utility function u(xn ) = xn , calculate the probability premium for the gamble G = (16, 4; ½, ½). 2.7 Using the same Bernoulli utility function, calculate the probability premium for the gamble G’ = (36, 16; ½, ½). 2.8 Calculate the coefficient of absolute risk aversion. Is it DARA, CARA or IARA? 2.9 Given the last 3 results, explain why the probability premium is less for gamble G than it is for gamble G’.

Question 3 3.1 Which of the following Bernoulli utility functions represent the same preferences?

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1) u(x) = ln(x)2) v(x) = 5ln(x) + 4

3) w(x) = ln(12x) + 4

4) z(x) = 2ln(x + 4)

Question 4 Consider the quadratic Bernoulli utility function u(w) = a + bw + cw2 4.1 What restrictions, if any, must be placed on the parameters a, b and c for this function to display risk aversion? 4.2 Over what domain of wealth can a quadratic Bernoulli utility function be defined, assuming decision makers’ preferences satisfy the assumption of monotonicity?

4.4 Show that this function, satisfying the restrictions in 4.1 and 4.2, cannot represent preferences that display DARA.

Bayesian Updating

For the following questions you will need to use the Total Probability Theorem and Bayes’ Rule/Theorem. I know this hasn’t been covered in class but you will have done this in the quants course and it’s important for the game theory that we’ll cover later. Total Probability Theorem: Given n mutually exclusive events A1,…,An whose probabilities sum to 1, then P(B) = P(B|A1)P(A1) + … + P(B|An)P(An), where B is an arbitrary event, and P(B|Ai) is the conditional probability of B assuming Ai. Bayes’ Theorem: P(A|B) = [P(B|A)P(A)]/P(B)

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4.3Given the gamble , show thatand that the probability premium

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Question 5 A test correctly identifies a disease D with probability 0.95 and wrongly diagnoses D with probability 0.01. From past experience, it is known that disease D occurs in a targeted population with frequency 0.2%. An individual is chosen at random from the targeted population and is given the test. Calculate the probability that: 5.1 The test is positive (+) [i.e. P(+)] 5.2 The individual actually suffers from disease D if the test turns out to be positive [i.e. P(D | +)] Question 6 You are a contestant in a game show. The host of the show says, “Behind one of those three doors is a new car, which is your prize should you choose the correct door”. You choose door A. The game show host then opens door B and shows you that there is nothing behind it. He then asks, “Now would you like to change your guess to door C?” Show that the answer is yes if the game show host opened a door he knew did not have a car behind it.