10.2.2 1.0 point possible (graded, results hidden) By how much money would his winnings need to increase or decrease so that Juan is indifferent between the $600 and the new game? (in case of an increase, insert a positive number; in case of a decrease, insert a negative number).
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- Suppose your friend Yvette offers you the following bet: She will flip a coin and pay you $3,000 if it lands heads up and collect $3,000 from you if it lands tails up. Currently, your level of wealth is $9,000. The graph shows your utility function from wealth. Use the graph to answer the following questions. UTILITY (Units of utility) 100 90 80 70 60 50 40 30 20 + 10 0 O 0 3 ** 6 с B 9 WEALTH (Thousands of dollars) 12 The shape of your utility function implies that you are a because the difference in utility between A and C is 15 ? individual, and, therefore, you the difference between C and accept the wager Which of the following best explain why the pain of losing $3,000 exceeds the pleasure of winning $3,000 for risk-averse people? Check all that apply. Risk-averse people overestimate the probability of losing money. Risk-averse people are relatively poor and cannot afford to lose any money. The more wealth that risk-averse people have, the less satisfaction they receive from an…Alex has a utility function U = W2, where W is his wealth in millions of dollars and U is the utility he obtains from that wealth. In the final stage of a game show, the host offers Alex a choice between (A) $9 million for sure, or (B) a gamble that pays $1 million with probability 0.4 and $16 million with probability 0.6. Use the blue curve (circle points) to graph Alex's utility function at wealth levels of $0, $1 million, $4 million, $9 million, and $16 million. Utility (Thousands) 5.0 4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0 0 2 4 6 8 10 12 14 Wealth (Millions of dollars) 16 18 20 V Utility Function (?)Jamal has a utility function U = W1/2, where W is his wealth in millions of dollars and U is the utility he obtains from that wealth. In the final stage of a game show, the host offers Jamal a choice between (A) $4 million for sure, or (B) a gamble that pays $1 million with probability 0.6 and $9 million with probability 0.4. a. b. c. d. Graph Jamal’s utility function. Is he risk averse? Explain. (2+2) Does A or B offer Jamal a higher expected prize? Explain your reasoning with appropriate calculations. (1) Does A or B offer Jamal a higher expected utility? Explain your reasoning with calculations. (2) Should Jamal pick A or B? Why?
- Zeynep is at the supermarket buying her bi-weekly groceries. As she arrived at the store right before closing, not much is left on the shelves; so her purchases are limited to beef, B, and an assortment of organic vegetables, V. Beef costs £50/kg and vegetables Ł10/kg. Her utility function is expressed as U(B,V) = B0-6v0.3. She has ±150 to spend. Please answer the following questions using this information: A) Given the vegetables cost less than beef, explain why Zeynep would not only buy vegetables. B) Write Zeynep's constrained maximization problem (objective function subject to her budget constraint). C) Using the Lagrangian technique, find how much beef and vegetables would Zeynep buy? D) Show that for the preferences represented as above, demand for beef is a function of only its own price, pg, and income, m, but not the price of vegetables, py. Note: Rather than using the given prices and income, use PB, Py and m. Find and comment on the comparative statics, i.e., how do changes…2. Alex and Bill share a flat. Alex enjoys reading in silence, while Bill enjoys listening to loud music. Bill controls his music system, which can produce noise levels up to 100 Decibels. Bill has no cash but Alex has £100. Their utility functions are given by: UA = 10(100 – D)i + (100 – M), %3D Ug = 10(D)i + M, where D denotes the Decibel level chosen by Bill, and M denotes any cash given to Bill by Alex. (a) Explain how Bill's music level affects Alex. Find the contract curve between Alex and Bill. Why is there a unique Pareto efficient noise level? (b) What is the maximum amount Alex would be willing to pay Bill to turn down the music to the Pareto efficient level? What is the minimum amount Bill would be willing to accept in order to turn down the music to the Pareto efficient level? Can the Pareto efficient noise level be achieved through private bargaining?Assume that someone has inherited 2,000 bottles of wine from a rich uncle. He or she intends to drink these bottles over the next 40 years. Suppose that this person’s utility function for wine is given by u(c(t)) = (c(t))0.5, where c(t) is each instant t consumption of bottles. Assume also this person discounts future consumption at the rate δ = 0.05. Hence this person’s goal is to maximize 0ʃ40 e–0.05tu(c(t))dt = 0ʃ40 e–0.05t(c(t))0.5dt. Let x(t) represent the number of bottle of wine remaining at time t, constrained by x(0) = 2,000, x(40) = 0 and dx(t)/dt = – c(t): the stock of remaining bottles at each instant t is decreased by the consumption of bottles at instant t. The current value Hamiltonian expression yields: H = e–0.05t(c(t))0.5 + λ(– c(t)) + x(t)(dλ/dt). This person’s wine consumption decreases at a continuous rate of ??? percent per year. The number of bottles being consumed in the 30th year is approximately ???
- You have k20per week to spend and two possible uses for this money,:telephoning friends back home and drinking coffee. Each hour of phoning costs k2 and each cup of coffee costs k1. Your utility functions U(X,Y)=XY,where X is the hours of phoning you do and Y the number of cups of coffee you drink. Now suppose the price of telephone calls drops to k1 per hour. What are your optimal choices? What is the resulting utility levelAntonio has a utility function U = W, where W is his wealth in millions of dollars and U is the utility he obtains from that wealth. In the final stage of a game show, the host offers Antonio a choice between (A) $9 million for sure, or (B) a gamble that pays $1 million with probability 0.4 and $16 million with probability 0.6. Use the blue curve (circle points) to graph Antonio's utility function at wealth levels of $0, $1 million, $4 million, $9 million, and $16 million. ? Utility (Thousands) 5.0 4.5 4.0 3.5 3.0 25 2.0 1.5 1.0 0.5 0 + 0 2 4 6 8 10 12 14 Wealth (Millions of dollars) 16 18 True or False: Antonio is risk averse. 20 20 Utility Function Choice True O False offers Antonio a higher expected prize. (Hint: The expected value of a random variable is the weighted average of the possible outcomes, where the probabilities are the weights.) Choice offers Antonio a higher expected utility. Antonio should pick choice,Nicolaus I Bernoulli offers his friend Pierre Rémond de Montmort a game where they need to repeatedly toss a fair ducat until they get a head for the first time. The game stops then, and they count the number n of coin tosses it took to get the desired outcome, and Montmort gets 2^n ducats. Assume that Montmort's utility function is u(w)=w^0.14. How much should Montmort pay to play this game?
- Suppose your classmate Shen offers you a wager: He will choose a playing card at random from a deck and pay you $1,000 if it is red, but you have to pay him $1,000 if it is black. Assume your wealth is currently $3,000. The graph shown below plots your utility as a function of wealth. Use the graph to answer the questions that follow. UTILITY (Units of utility) 100 90 80 70 60 50 40 30 20 10 0 0 1 B +++ A 2 * 3 WEALTH (Thousands of dollars) The shape of your utility function implies that you are a the difference in utility between C and A is less than 4 5 ? risk-averse individual, and, therefore, you would not the difference between A and B. accept the wager because Which of the following sentences most appropriately describe why the pain of losing $1,000 is greater than the joy of winning $1,000 for individuals who are risk averse? Check all that apply. Risk-averse people are relatively wealthy and simply do not need the additional money. ✔ The utility function of a risk-averse person…Oliver takes $2500 with him to a camp and there is 50% chance he will lose $900 on his way. Suppose Oliver can buy an insurance policy that will totally cover his loss, what maximal amount will he be willing to pay for such insurance? Oliver’s utility function is given by the function U(E) = E0.5 where E is the amount that he spends on the camp without any saving. a. $325 b. $475 c. $650 d. $5355.3 (0) Ambrose, the nut and berry consumer, has a utility function U(21,x2) = 4/T +x2, where ¤; is his consumption of nuts and r2 is his consumption of berries. %3D (a) The commodity bundle (25,0) gives Ambrose a utility of 20. Other points that give him the same utility are (16, 4), (9, ), (4, - ), (1, –), and (0, ). Plot these points on the axes below and draw a red indifference curve through them. (b) Suppose that the price of a unit of nuts is 1, the price of a unit of berries is 2, and Ambrose's income is 24. Draw Ambrose's budget line with blue ink. How many units of nuts does he choose to buy? (c) How many units of berries? (d) Find some points on the indifference curve that gives him a utility of 25 and sketch this indifference curve (in red). (e) Now suppose that the prices are as before, but Ambrose's income is 34. Draw his new budget line (with pencil). How many units of nuts will he choose? How many units of berries?