2. The utility function is u(x, y) = min{x,y}. Suppose that 1) with probability 0.5, (I, Px, Py) = (4,1,1) and 2) with probability 0.5, (I, Px, Py) = (12,6,2). Explain graphically whether an increase in x increases or decreases the riskiness of the utility gamble. 21
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- 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…3. The utility function is u(x, y) = √x+y. Suppose that 1) with probability 0.5, (I, Px, Py) (8, 2, 2) and 2) with probability 0.5, (I, Px, Py) = (8, 4, 8). Explain graphically whether an increase in x increases or decreases the riskiness of the utility gamble. U 6 5 4 3 2 1 0 0 0.25 0.5 0.75 1 1 1.25 1.5 1.75 28The investor is considering how to optimally invest 1000 euros in stocks and bonds. Let's assume that the optimal decision is made based on expected utility. Suppose the investor has a utility function u(x)=ln(1+x), where x is their wealth. Let y be the proportion invested in stocks and 1−y be the proportion invested in bonds. By investing in stocks, the investor earns 1% with a probability of 39.5% and 4% with a probability of 60.5%. By investing in bonds, the investor earns a certain 2.8%. What proportion of the investment will the investor allocate to stocks and what proportion to bonds?
- A lottery has a grand prize of $1,000,000, 2 runner-up prizes of $100,000 each, 6 third-place prizes of $10,000 each, and 19 consolation prizes of $1,000 each. If a 4 million tickets are sold for $1 each, and the probability of any ticket winning is the same as that of any other winning, find the expected return on a $1 ticket. (Round your answer to 2 decimal places.(Fill in the 4 blanks.) A monthly pass for the Stockholm subway costs $100, and fare dodgers who are caught face a fine of about $160. Stockholm transport has increased the number of ticket checks conducted, and the probability of being caught while riding the subway without paying is now 1%. Liam rides the subway 50 times a month, so that each month the probability that he is caught equals 50 x 1% = !3! 50% and he expects to pay 50 (0.01 x $160) = $80 in fines each month he rides the subway without buying the monthly pass. If Liam does not purchase a monthly pass, his expected monthly savings are $ (accepted format is dollars and cents xXx.xx). Liam's utility of saving $S equals VS , so that if he does not purchase a monthly pass, his expected utility equals (round to 2 decimals: xxx.xx). Liam's certainty equivalent is savings equal to $52.42, and thus Liam's (do not capitalize your answer and make sure to spell correctly) is $7.58. Liam can purchase full insurance from Planka.nu,…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.
- An investor with capital x can invest any amount between0 and x; if y is invested then y is eitherwon or lost, with respectiveprobabilities p and 1− p. If p > 1/2, how much should be invested byan investor having a exponential utility function u(x) = 1 − e −bx ,b > 0.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.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?
- 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. What is expected gains/loss.Eric has a job at an electronics store in a mall. Eric doesn't like to work hard, and it costs him $100 to do so. Eric's employer cannot observe whether Eric works hard or not. If Eric works hard, there is a 75% probability that electronics goods profits will equal $400 a day and a 25% probability that electronics goods profits will equal $100 a day. If Eric shirks, there is a 75% probability that electronics goods profits will equal S100 a day and a 25% probability that electronics goods profits will equal $400 a day. Suppose Eric is paid $200 if electronics goods profits are $400 a day and $50 if electronics goods profits are S100 a day. Eric will because the net gain of from shirking is than the net gain of from working hard. O shirk; $87.50; more; $62.50 O shirk; $125; more; $118 O work hard; S50; less; $62.50 O work hard; $100; less; $250Players would draw a card from a standard 52 card deck. Whatever card they drew determined what they won. If they draw a face card (Jack, King, Queen) then they win $5. If they draw an Ace, they win $15. For all other cards, they win nothing. A. Fill out the probability distribution table with the probabilities of each possible outcome for this game. Round decimals to four places. x $15 $5 $0 P(x) B. What is the expected value of the distribution above? (Round to the nearest cent, two decimal places.) C. If players were charged $2 per game, would they make and average profit on the games over time, or would they take an average loss over time? D. If players were charged $3 per game, would they make and average profit on the games over time, or would they take an average loss over time?
