1-7 There is one period. Assume a representative agent with utility function U(c) = . In parts c) through i) assume the following: • 7 = 2 and 6 = .94. • Consumption at t = 0 is Co = 25. • At t = 1 one of two states 01 and 02 eventuate with probability 7₁ = respectively. . There are two complex securities s¹ and s². • s¹ has a payoff of 23 in 01 and 27 in 02. s² has a payoff of 21 in 0₁ and 32 in 02. .6, and T2 = .4,
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(b) Derive an expression for the stochastic discount factor mt+1?
(c)What are the prices q0s1and q0s2 of complex securities s1 and s2 at t = 0?
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- Gary likes to gamble. Donna offers to bet him $31 on the outcome of a boat race. If Gary's boat wins, Donna would give him $31. If Gary's boat does not win, Gary would give her $31. Gary's utility function is p1x^21+p2x^22, where P₁ and p2 are the probabilities of events 1 and 2 and where x₁ and x₂ are his wealth if events 1 and 2 occur respectively. Gary's total wealth is currently only $80 and he believes that the probability that he will win the race is 0.3. Which of the following is correct? (please submit the number corresponding to the correct answer). 1. Taking the bet would reduce his expected utility. 2. Taking the bet would leave his expected utility unchanged. 3. Taking the bet would increase his expected utility. 4. There is not enough information to determine whether taking the bet would increase or decrease his expected utility. 5. The information given in the problem is self-contradictory.Gary likes to gamble. Donna offers to bet him $31 on the outcome of a boat race. If Gary’s boat wins, Donna would give him $31. If Gary’s boat does not win, Gary would give her $31. Gary’s utility function is p1x^21+p2x^22, where p1 and p2 are the probabilities of events 1 and 2 and where x1 and x2 are his wealth if events 1 and 2 occur respectively. Gary’s total wealth is currently only $80 and he believes that the probability that he will win the race is 0.3. Which of the following is correct? (please submit the number corresponding to the correct answer). Taking the bet would reduce his expected utility. Taking the bet would leave his expected utility unchanged. Taking the bet would increase his expected utility. There is not enough information to determine whether taking the bet would increase or decrease his expected utility. The information given in the problem is self-contradictory.Y5 Alfred is a risk-averse person with $100 in monetary wealth and owns a house worth $300, for total wealth of $400. The probability that his house is destroyed by fire (equivalent to a loss of $300) is pne = 0.5. If he exerts an effort level e = 0.3 to keep his house safe, the probability falls to pe = 0.2. His utility function is: U = w0.5 – e where e is effort level exerted (zero in the case of no effort and 0.3 in the case of effort).a. In the absence of insurance, does Alfred exert effort to lower the probability of fire?HINT: Calculate and compare the expected utility i) with effort, and ii) without effort. If effort is exerted, then the effort cost is paid regardless of whether or not a fire occurs.b. Alfred is considering buying fire insurance. The insurance agent explains that a home owner’s insurance policy would require paying a premium α and would repay the value of the house in the event of fire, minus a deductible “D”. [A deductible is an amount of money that the…
- Your utility function is given by M1/2. You have $100 and are planning to invest in a venture where you can win or lose 50 with equal probability. Will you accept the venture? What is the minimum gain you need to make in the good scenario such that you will invest in the venture?Arielle is a risk-averse traveler who is planning a trip to Canada. She is planning on carrying $400 in her backpack. Walking the streets of Canada, however, can be dangerous and there is some chance that she will have her backpack stolen. If she is only carrying cash and her backpack is stolen, she will have no money ($0). The probability that her backpack is stolen is 1/5. Finally assume that her preferences over money can be represented by the utility function U(x)=(x)^0.5 Suppose that she has the option to buy traveler’s checks. If her backpack is stolen and she is carrying traveler’s checks then she can have those checks replaced at no cost. National Express charges a fee of $p per $1 traveler’s check. In other words, the price of a $1 traveler’s check is $(1+p). If the purchase of traveler’s checks is a fair bet, then we know that the purchase of traveler checks will not change her expected income. Show that if the purchase is a fair bet, then the price (1+p) = $1.25.Suppose that an individual is just willing to accept a gamble to win or lose $1000 if the probability ofwinning is 0.6. Suppose that the utility gained if the individual wins is 100 utils. How much utility does one lose if one loses the gamble?
- Consider an individual who gets a utility of u(x) - x^1/2 from his total wealth x. Amsume that he has 160.000 AZN in the bank and owns a car with a value of 90,000 AZN. It is expected that will be stolen within the next year with 20% probability, whereas nothing will happen with. Your company tries to sell him an insurance package with the following properties; as an insurance premium now. (ii) if his car is stolen, your company will pay him a partial ation of 55,000 AZN. (iii) if his car is not stolen, there will be no paytent made by your .Should the individund buy this package, if the insurance premium in 12,500 AZN? ExplainQ1. A farmer believes there is a 50-50 chance that the next growing season will be abnormally rainy. His expected utility function has the form Expected utility = 0.5lnYNR + 0.5lnYR Where and represent the farmers income in the state of ‘normal rain’ and ‘rainy’ respectively. Suppose the farmer must choose between two crops that promise the following income prospects Crop YNR YR Wheat $83,000 $10,000 Maize $83,000 $15000 What mix of wheat and maize would provide maximum expected utility to this farmer?An investor is confronted with three investment opportunities as indicated in the following table: Investment A probability 1/2 1/4 1/4 Investment B Investment C probability 1/4 1/2 1/4 probability 1/2 3/8 1/8 return return return -$15 -$10 $0 $20 $10 $10 $35 $20 $20 Please answer Questions 10-13 based on this information. Consider Investment C above and an agent with a utility function U(x) = 1 – e-*. Assume that the initial wealth is 0. Based on the Principle of Expected Utility, how much is this agent willing to pay to receive the expected return of this investment without bearing the risk? O 6.25 0.5
- A risk-averse agent, Andy, has power utility of consumption with riskaversion coefficient γ = 0.5. While standing in line at the conveniencestore, Andy hears that the odds of winning the jackpot in a new statelottery game are 1 in 250. A lottery ticket costs $1. Assume his income isIt = $100. You can assume that there is only one jackpot prize awarded,and there is no chance it will be shared with another player. The lotterywill be drawn shortly after Andy buys the ticket, so you can ignore therole of discounting for time value. For simplicity, assume that ct+1 = 100even if Andy buys the ticket How large would the jackpot have to be in order for Andy to play thelottery? b) What is the fair (expected) value of the lottery with the jackpot youfound in (a)? What is the dollar amount of the risk premium that Andyrequires to play the lottery? Solve for the optimal number of lottery tickets that Andy would buyif the jackpot value were $10,000 (the ticket price, the odds of winning,and Andy’s…Consider the lottery that assigns a probability r of obtaining a level of consumption CH and a probability 1-T of obtaining a low level of consumption cL an individual facing such a lottery with utility function u(c) that has the properties that more is better (that is, a strictly positive marginal utility of consumption at all levels of c) and diminishing marginal utility of consumption, u"(c) CL. Consider du(c) for the first derivative of the utility function with respect to dc d²u(c) dc2 du' (c) consumption and u"(c) which is also the derivative of the first derivative of the utility function). to be the second derivative of the utility function dcMicroeconomics Wilfred’s expected utility function is px1^0.5+(1−p)x2^0.5, where p is the probability that he consumes x1 and 1 - p is the probability that he consumes x2. Wilfred is offered a choice between getting a sure payment of $Z or a lottery in which he receives $2500 with probability p = 0.4 and $3700 with probability 1 - p. Wilfred will choose the sure payment if Z > CE and the lottery if Z < CE, where the value of CE is equal to ___ (please round your final answer to two decimal places if necessary)