Consider two lotteries L1 = {200, 0.20; 400, 0.50; 500, 0.30} L2 = {500, 0.10; 300, 0.70; 100, 0.20} (a) Check for first-order stochastic dominance between these lotteries. (b) Consider an expected utility maximiser with the utility function u(x) = ln x. Which lottery is going to be preferred? What is the risk premium (RP) of the lottery L1? (c) Consider a decision maker who follows the modified expected utility approach (MEU). Assume the utility function to be u(x) = ln x and the probability weight function to be w(p) = p^0.50/{p^0.50 + (1 − p)^0.50}^2 Which lottery is going to be preferred? (d) Consider a decision maker who follows the rank dependent utility approach (RDU). Assume the utility function to be u(x) = ln x and the probability weight function to be w(p) = p^0.50/{p^0.50 + (1 − p)^0.50}^2 Which lottery is going to be preferred?
Question
Consider two lotteries L1 = {200, 0.20; 400, 0.50; 500, 0.30} L2 = {500, 0.10; 300, 0.70; 100, 0.20} (a) Check for first-order stochastic dominance between these lotteries. (b) Consider an expected utility maximiser with the utility function u(x) = ln x. Which lottery is going to be preferred? What is the risk premium (RP) of the lottery L1? (c) Consider a decision maker who follows the modified expected utility approach (MEU). Assume the utility function to be u(x) = ln x and the probability weight function to be w(p) = p^0.50/{p^0.50 + (1 − p)^0.50}^2 Which lottery is going to be preferred? (d) Consider a decision maker who follows the rank dependent utility approach (RDU). Assume the utility function to be u(x) = ln x and the probability weight function to be w(p) = p^0.50/{p^0.50 + (1 − p)^0.50}^2 Which lottery is going to be preferred?
Solution
(a) First-order stochastic dominance (FOSD) occurs when one lottery is preferred to another because it offers higher payoffs with the same or higher probabilities. To check for FOSD, we compare the cumulative distribution functions (CDFs) of the two lotteries.
For L1, the CDF is: P(X ≤ 200) = 0.20 P(X ≤ 400) = 0.20 + 0.50 = 0.70 P(X ≤ 500) = 0.20 + 0.50 + 0.30 = 1.00
For L2, the CDF is: P(X ≤ 100) = 0.20 P(X ≤ 300) = 0.20 + 0.70 = 0.90 P(X ≤ 500) = 0.20 + 0.70 + 0.10 = 1.00
Comparing these CDFs, we see that for all x, P(X ≤ x) for L1 is not less than P(X ≤ x) for L2, and there exists an x for which P(X ≤ x) for L1 is strictly greater than P(X ≤ x) for L2. Therefore, L1 first-order stochastically dominates L2.
(b) An expected utility maximiser with utility function u(x) = ln x will prefer the lottery with the higher expected utility. The expected utility of a lottery is the sum of the utilities of its outcomes, each weighted by its probability.
For L1, the expected utility is: E[U(L1)] = 0.20ln(200) + 0.50ln(400) + 0.30*ln(500)
For L2, the expected utility is: E[U(L2)] = 0.10ln(500) + 0.70ln(300) + 0.20*ln(100)
Calculate these values to determine which lottery is preferred.
The risk premium of L1 is the amount of money that the expected utility maximiser would be willing to pay to avoid the risk of the lottery. It is the difference between the expected value of the lottery and the certain equivalent, which is the amount of money that gives the same utility as the expected utility of the lottery.
E[L1] = 0.20200 + 0.50400 + 0.30*500 CE = exp(E[U(L1)]) RP = E[L1] - CE
(c) The modified expected utility (MEU) approach involves weighting the probabilities of the outcomes by a probability weight function. The decision maker will prefer the lottery with the higher MEU.
For L1, the MEU is: MEU(L1) = w(0.20)*ln(200) + w(0.50)*ln(400) + w(0.30)*ln(500)
For L2, the MEU is: MEU(L2) = w(0.10)*ln(500) + w(0.70)*ln(300) + w(0.20)*ln(100)
Calculate these values to determine which lottery is preferred.
(d) The rank dependent utility (RDU) approach is similar to the MEU approach, but the probabilities are weighted differently. The decision maker will prefer the lottery with the higher RDU.
For L1, the RDU is: RDU(L1) = w(0.20)*ln(200) + w(0.70)*ln(400) + w(1.00)*ln(500)
For L2, the RDU is: RDU(L2) = w(0.20)*ln(100) + w(0.90)*ln(300) + w(1.00)*ln(500)
Calculate these values to determine which lottery is preferred.
Similar Questions
Consider an expected utility maximiser with utility function u(w) = w^a, a > 0. Verify whether this individual always rejects the lottery {(w + 110), 0.5; (w −100), 0.5} for all wealth level. note - answer this question completely in detail, do not leave any calculations incomplete
A.1 If an individual is risk averse, then:i. He prefers any lottery to amounts of money that are certain.ii. The certainty equivalent of a lottery is higher than its expected value.X. The utility function is concave.iv. The utility function is decreasing.
Consider an individual who has the following value function: v(x) = x^1/2 for gains and v(x) = −2(|x|)^1/2 for losses. This individual is facing a lottery which consist of winning Rs. 25000 with a 25 % chance, Rs. 15000 with a 25 % chance, and losing Rs. 5000 with a 50 % chance. Suppose the probability weight function is p^2. What will be the expected utility of this lottery? What will be the value of this lottery as per the RDU approach? What will be the certainty equivalent of this lottery as per the expected theory approach? What will be the certainty equivalent of this lottery as per the RDU approach?
Consider two random variables (lotteries) ˜ω1 and ˜ω2. Let ˜ω2 be dominated by ˜ω1 in the senseof the rst-degree stochastic dominance. In class we argued that the expected utility of ˜ω2 isstrictly less than of ˜ω1, i.e. E[u(˜ω2)] − E[u(˜ω1)] < 0 for any increasing utility function u(w). Explain why expected payo of ˜ω2 is also necessarily less than of ˜ω1, i.e. whyE[˜ω2] < E[˜ω1]. Give an example showing that the reverse is not true. That is, give an example of tworandom variables, such that one random variable gives a better expected payo than theother, i.e. E[˜ω1] > E[˜ω2], but ˜ω2 is NOT rst-degree stochastically dominated by ˜ω1.2
Consider an individual who has the following value function: v ( x )=√(x /2) for gains and v ( x )=-2 √|x| for losses. This individual is facing a lottery which consist of winning Rs. 1600 with a 25% chance, Rs. 400 with a 25% chance, and winning nothing with a 50% chance. The individual counts all outcomes as losses; that is, he takes as his reference point the best possible outcome under consideration. Suppose the weight function is p^2. What will be the expected value/utility of this lottery as per the prospect theory approach? What will be the expected value/utility of this lottery as per the RDU approach? What will be the certainty equivalent of this lottery as per the prospect theory approach?
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.