4. $1000 with of $325. (a) (b) Adam is risk averse. He is offered a choice between a gamble that pays a probability of 25% and $100 with a probability of 75%, or a sure payment What is the expected payment of the gamble? Will Adam prefer gamble over sure payment? Would he change his mind if the sure payment is $320 instead of $325? (c) If this individual has a utility function u(x) = lnx, would he prefer the payment of $320 or the gamble? (d) In the CRRA utility family u(x) = x¹7. If this individual has a 0.01, would he prefer the payment of $320 or the gamble? utility where y =
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In the CRRA utility family u(x) = x^(1−γ). If this individual has a
utility where γ = 0.01, would he prefer the payment of $320 or the gamble?
- Economics Shawn's consumption is subject to risk. With probability 0.75 he will enjoy 10000 in consumption, but with probability 0.25 he will have only 3600. His utility function for consumption is given by v(c) = Vc. -What is the expected value of Shawn's consumption? -What is his expected utility? -What is his certainty equivalent of having 10000 with probability 0.75 and 3600 with probability 0.25?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…Johnny is "paid" by his parents $2o if he gets a grade A, $10 if he gets a grade B, whereas he has to pay his parents back $5 if he gets a grade other than A or B. On average 20% of the grades he gets are A, and 30% are grades B. What is the expected value of what he "earns" per grade ? What is the expected value of what he "earns" at school weekly if on average he gets five grades a week ? How long should Jim save until he collects enough money to buy a pair of brand new Hi-Fi headphones that cost $225?
- 2 Consider the two investments listed below with possible outcomes and probabilities: INVESTMENT (in $1000) SAFE RISKY INVESTMENT AMOUNTⓇ 40+ 40+ GOOD SCENARIO OUTCOME 45+ 80+ AVERAGE+ SCENARIO PROB OUTCOME 0.40* 0.40€ 42+ 45+ BAD+ SCENARIO PROB OUTCOME PROB 0.20 35+ 0.20 10+ 0.40€ 0.40+ b) a) Suppose I have utility function U(*) = (x)2. What is the expected utility from each investment? Which investment will I choose, if any? Show and explain your work and provide the intuition. c) What is the value of the risk premium for the SAFE investment? Show and explain your work and provide the intuition. d) What is the value of the risk premium for the RISKY investment? Show and explain your work and provide the intuition.< +1. A woman with current wealth X has the opportunity to bet an amount on the o ccurrence of an event that she knows will occur with probability P. If she wager s W, she will received 2W, if the event occur and if it does not. Assume that t he Bernoulli utility function takes the form u(x) = -e-TX with r> 0. How much should she wager? Does her utility function exhibit CARA, DARA, IARA?When playing roulette at a casino, a gambler is trying to decide whether to bet $10 on the number 30 or to bet $10 that the outcome is any one of the three possibilities 00, 0, or 1. 3 The gambler knows that the expected value of the $10 bet for a single number is - 53¢. For the $10 bet that the outcome is 00, 0, or 1, there is a probability of 38 of making a net profit of $30 and a probability of losing $10. 35 38 a. Find the expected value for the $10 bet that the outcome is 00, 0, or 1. b. Which bet is better: a $10 bet on the number 30 or a $10 bet that the outcome is any one of the numbers 00, 0, or 1? Why? a. The expected value is $. (Round to the nearest cent as needed.) b. Since the expected value of the bet on the number 30 is C than the expected value for the bet that the outcome is 00, 0, or 1, the bet on is better.
- 2. Kier, in The scenario, wants to determine how each of the 3 companies will decide on possible new investments. He was able to determine the new investment pay off for each of the three choices as well as the probability of the two types of market. If a company will launch product 1, it will gain 50,000 if the market is successful and lose 50,000 if the market is a failure. If a company will launch product 2, it will gain 25,000 if the market is successful and lose 25,000 if the market will fail. If a company decides not to launch any of the product, it will not be affected whether the market will succeed or fail. There is a 56% probability that the market will succeed and 44% probability that the market will fail. What will be the companies decision based on EMV? What is the decision of each company based on expected utility value?8 An investor with initial wealth $20000 and utility function U(x) = ln(x) is considering an investment that has a 80% chance of gaining r% and a 20% chance of losing s%. (1) Find in terms of r and s the certainty equivalent of this investment. (2) If s = 10, find the range of values of r for which the investor will avoid this investment.,4. Consider the situation in which player 1 knows what game is played (the first or the second). But player 2 only knows that the first game is played with probability 1/3 and the second game is played with probability 2/3. Player 1 Player 1 Player 2 S B S 1,1 0,0 B 0,0 0,0 Player 2 S B S 0,0 0,0 B 0,0 2,2 (a) Describe the game as a Bayesian game. (b) Find the Bayesian Nash equilibria.
- 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.Becky is deciding whether to purchase an insurance for her home againtst burglary. the payoff for her is shown as follow: Net worth of her Net worth of her home: $ 20000 burglary(10%) Net worth of her Net worth of her home: $50000 burglary (90%) The insueance would cover all the loss from burlary and the insurance fee is $8000. Her utility funtion is given as u=w ^0.3 Should Beck purchase the insurance Explain.7. Suppose the only game in town involves flipping a fair coin (so Heads and Tails are equally likely), with a $x bet. If Heads comes up, the payoff is $0.9x; if Tails comes up, you lose the $x. You have $10,000, and must win at least $5,000 by tomorrow morning to pay off a debt to a mean dude. a. Compute the likelihood of winning at least $5000 by making a single bet of $10,000. b. Compute the likelihood of winning at least $1000 by playing the game 10,0000 times and betting a dollar each time. What is the likelihood of not losing money? Message learned?