2. If h(y) denotes an individual's happiness (called “utility" in economics) when having income Y, then yh" (y) h'(y) R(y) is called the coefficient of relative risk aversion, which measures how much an indi- vidual is willing to take risks. Compute R for the following utility functions, where a, b, c, d are constants ay + 6 h(y) = (ay + b)°, h(y) cy +d° In the computations, you can assume that no denominator is 0. Simplify as much as possible.
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- 1. A standard model of choice under risk is Expected Utility Theory (EUT) in which preferences over lotteries that pay monetary prizes (x₁, x2, ..., xs) with probabilities (P1, P2, ..., Ps) with Eps = 1 are represented by the function L S (a) What does it mean to say that a function represents the consumer's prefer- ences? Σpsu(xs) Choice 1 8=1 (b) State and briefly comment on the axioms required for the EUT representation. (c) Consider the following experiment of decision making under risk in which sub- jects are asked which lottery they prefer in each of the following two choices: Lottery B 0 with prob. 0.01 10 with prob. 0.89 50 with prob. 0.10 Lottery D Choice 2 Lottery A 0 with prob. 0 10 with prob. 1 50 with prob. 0 Lottery C 0 with prob. 0.90 10 with prob. 0 50 with prob. 0.10 Suppose that the modal responses are Lottery A in Choice 1 and Lottery D in Choice 2. Assume that utility of zero is equal to zero and illustrate why it is not possible to reconcile these experimental…1. Now, imagine that Port Chester decides to crack down on motorists who park illegally by increasing the number of officers issuing parking tickets (thus, raising the probability of a ticket). If the cost of a ticket is $100, and the opportunity cost for the average driver of searching for parking is $12, which of the following probabilities would make the average person stop parking illegally? Assume that people will not park illegally if the expected value of doing so is negative. Check all that apply. A. 9% B. 18% C. 17% D. 10% 2. Alternatively, the city could hold the number of officers constant and discourage parking violations by raising the fine for illegal parking. Suppose the average probability of getting caught for parking illegally is currently 10% citywide, and the average opportunity cost of parking is, again, $12. The fine that would make the average person indifferent between searching for parking and parking illegally is ____ , assuming that people will not…3. Further questions Now, imagine that Port Chester decides to crack down on motorists who park illegally by increasing the number of officers issuing parking tickets (thus, raising the probability of a ticket). If the cost of a ticket is $100, and the opportunity cost for the average driver of searching for parking is $12, which of the following probabilities would make the average person stop parking illegally? Assume that people will not park illegally if the expected value of doing so is negative. Check all that apply. 13% 21% 9% 10% Alternatively, the city could hold the number of officers constant and discourage parking violations by raising the fine for illegal parking. Suppose the average probability of getting caught for parking illegally is currently 10% citywide, and the average opportunity cost of parking is, again, $12. The fine that would make the average person indifferent between searching for parking and parking illegally is 5 park illegally if the expected value of…
- 2. Alice believes that her car would cost £12500 to replace if it was stolen or damaged. Based on crime statistics for the area she lives in, she believes that the probability of her car being stolen or damaged is 0.15. (i) Alice's utility function is given by U(w) = ln(w) for w > 0 and she as £35000 in the bank. Calculate how much Alice would be prepared to pay (in a single payment) to insure her car against theft or damage (ii) Repeat the calculation in the previous part but now assume Alice has £500000 in the bank.Utility Theory You live in an area that has a possibility of incurring a massive earthquake, so you are considering buyingearthquake insurance on your home at an annual cost of $180. The probability of an earthquake damagingyour home during one year is 0.001. If this happens, you estimate that the cost of the damage (fully coveredby earthquake insurance) will be $160,000. Your total assets (including your home) are worth $250,000. A. Apply Bayes’ decision rule to determine which alternative (take the insurance or not) maximizes yourexpected assets after one year.1. Mr. Smith can cause an accident, which entails a monetary loss of $1000 to Ms. Adams. The likelihood of the accident depends on the precaution decisions by both individuals. Specifically, each individual can choose either "low" or "high" precaution, with the low precaution requiring no cost and the high precaution requiring the effort cost of $200 to the individual who chooses the high precaution. The following table describes the probability of an accident for each combination of the precaution choices by the two individuals. Adams chooses low precaution Adams chooses high precaution Smith chooses low precaution Smith chooses high precaution 0.8 0.5 0.7 0.1 1) What is the socially efficient outcome? For each of the following tort rules, (i) construct a table describing the individuals' payoffs under different precaution pairs and (ii) find the equilibrium precaution choices by the individuals. 2) a) No liability b) Strict liability (with full compensation) c) Negligence rule (with…
- 4. Kate has von Neumann-Morgenstern utility function U(x1,x2) = m7. She currently has $2025. a. Would she be willing to undertake a gamble that involves a gain $2875 with probability + and a loss of $1125 with probability ? Show your work and explain your answer. b. Would she be willing to undertake a gamble that involves a gain $2599 with probability and a loss of $800 with probability ? Show your work and explain your answer.2. Suppose you asked the following question to Person A and Person B: "How much are you willing to pay to avoid the following fair gamble – win $100 with 50% chance and lose $100 with 50% chance (thus, Variance is equal to 10,000)?" A's answer- $2 B's answer-$10 Assuming that A and B have CARA utility function, a) compute their absolute risk aversion coefficients (approximately) and b) compute their risk premiums for avoiding the following new gamble - win $500 with 50% chance and lose $500 with 50% chance.1. A dealer decides to sell a rare book by means of an English auction with a reservation price of 54. There are two bidders. The dealer believes that there are only three possible values, 90, 54, and 45, that each bidder’s willingness to pay might take. Each bidder has a probability of 1/3 of having each of these willingnesses to pay, and the probabilities for each of the two bidders are independent of the other’s valuation. Assuming that the two bidders bid rationally and do not collude, the dealer’s expected revenue is approximately ______. 2. A seller knows that there are two bidders for the object he is selling. He believes that with probability 1/2, one has a buyer value of 5 and the other has a buyer value of 10 and with probability 1/2, one has a buyer value of 8 and the other has a buyer value of 15. He knows that bidders will want to buy the object so long as they can get it for their buyer value or less. He sells it in an English auction with a reserve price which he must…
- 2. Consider a cheap talk game in which Nature moves by choosing a sender's type, where the type space has four elements: −1, 1, 2, and 3, each occur- ring with equal probability of 1½. The sender learns his type and chooses one of three possible messages: bumpy, smooth, and slick. The receiver observes the sender's message and then chooses one of three actions: 0, 5, and 10. The sender's payoff equals his type multiplied by the receiver's action. The receiver's payoff equals the sender's type multiplied by twice the receiver's payoff. a. Find a separating perfect Bayes-Nash equilibrium. b. Find a semiseparating perfect Bayes-Nash equilibrium.1. Show if the following utility functions represent risk averse, risk neutral or risk loving preferences. u(c) = 10° + 3 u(c) = C2 + 3C i. ii. ii. u(c) = e4C iv. u(c) = 1 – e-C4.6. A person purchases a dozen eggs and must take them home. Although making trips home is costless, there is a 50 percent chance that all of the eggs carried on one trip will be broken during the trip. This person con- siders two strategies: Strategy 1: Take the dozen eggs in one trip. Strategy 2: Make two trips, taking six eggs in each trip. a. List the possible outcomes of each strategy and the probabilities of these outcomes. Show that, on aver- age, six eggs make it home under either strategy. b. Develop a graph to show the utility obtainable under each strategy. c. Could utility be improved further by taking more than two trips? How would the desirability of this possibility be affected if additional trips were costly?