Assume you have the following von Neumann-Morgenstern utility function: u(w)=10000-(200-w/1000)^2 with w denoting wealth Work out the first-order condition
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Assume you have the following von Neumann-Morgenstern utility function:
u(w)=10000-(200-w/1000)^2 with w denoting wealth
Work out the first-order condition
Step by step
Solved in 2 steps
- 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…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 ???Answer the following questions using the following information. Columns 1 and 2 in the table below show the marginal utility that Cody gets by purchasing products A and B. Column 3 shows the marginal utility Cody gets from saving Assume that the price of A is $13 the price of B is $10, and Cody has an income of $129. a) Find the following series of MU/$ for each column. Note: Keep as much precision as possible during your calculations. Your final answer should be accurate to at least two decimal places Column 1 Column 2 Column 3 Units of A MU MU/S Units of B MU MU/$ Number of $ saved MU MU/S 68 6.8 1 80 6.15 1. 13 0. 2 71 5.46 2 63 6.3 2 10 0. 3 66 5.08 3 56 5.6 3 7 4 61 4.69 4. 46 46 0. 54 4.15 40 4 45 3.46 31 3.1 6. 1 39 3 22 2.2 0. 29 2.23 8 17 1.7 8 b) What quantities of A and B will Cody purchase in maximizing his utility? Quantity of A: 0 Quantity of B: 0 c) How many dollars will Cody save? Dollars Saved = $0 SAVE AND CLOSE
- V*(p1, p2, 1) = True False P₁ P₂ 1² is a valid indirect utility function.Suppose an individual in the Grossman model is trying to decide what to have for dinner. His options are as below. Each dish has an effect on the level of home good Z and health H. Мeal Home good Z Нeath H Steak and eggs (A) Kale salad with broccoli (B) Entire box of cookies (C) +7 -2 -2 +5 +10 -20 Suppose the dinner's single-period utility function is U=3Z+H • If the individual is trying to maximize his single-period utility, and he can only select one item from the table (assuming he can afford any item in the table). Which meal would he choose? Please type in A, B, or C (do not enter space, punctuation, or any other symbols or words) • A miracle pill is discovered that halves the negative health impact of cookies. How does this impact the individual's choice? What meal would be chosen now? Please type in A, B, or C (do not enter space, punctuation, or any other symbols or words) • If the individual lives in multi-period rather than single-period, would he value Z or H more in…Violet buys pies (x) and champagne (y) with her income of $400 and her utility function over pies (x) and champagne (y) (assumed to be divisible goods, where any real number unit pie or champagne is feasible) is Cobb-Douglas and given by u(x, y) = xy. The price of champagne is pY = $10 per bottle. (a) Pies cost pX = $10 per pie if she buys between zero and 20 pies; if she buys more than 20 pies, each additional pie is half-price, i.e., pX = $5. Draw her budget set, carefully labelling all relevant points. calculate the quantities of pies and champagne she will consume when she maximizes her preferences. (b) (HARDER! We haven’t covered this scenario in class) Pies cost pX = $10 per pie if she buys between zero and 20 pies; if she buys more than 20 pies, each pie purchased is half-price, i.e., pX = $5. The discount applies to all units purchased, not the additional ones! Draw her budget set, carefully labelling all relevant points. Calculate the quantities of pies and champagne she will…
- Suppose Jimi has reference dependent preferences over guitars and money as in Tversky and Kahneman (1991). His utility functions are given below. Gains Gains 400 2 -2 -2 Guitars 24 Losses Losses -600 -2 What is the least amount of money Jimi is willing to accept to sell one of his guitars? (just enter a dollar amount, i.e., "1000", not "$1000"A consumer has the following utility function (shown in image) where ? is the number of spa days and ? is the number of city breaks consumed. Suppose that the price of a spa day is £200 and the price of a city break is £300. (i) Set up the economic problem and find the numbers of spa days and city breaks that minimise expenditure if 12,800 units of utility are to be obtained.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?
- Eren’s two main hobbies are taking vacations overseas (V) and eating expensive meals (M). His utility function is given as: U(V,M) = V^2MLast year, the average price of taking a vacation overseas was US$200 and the average price of an expensive meal is $50. However, due to supply problems in Onions, the average price of an expensive meal rose to $75. The average price of a vacation did not change. His income, which is $1500, did not change. Suppose that the Department of Welfare wants to know how much should be given to Eren to offset his change un utility due to the price increase of an expensive meal. Calculate the compensative variation (CV).You are considering going to a football game. However, the roads are cover in ice due to bad weather. Your ticket was a gift. You derive a value of z from attending the game, and a cost of D for driving on the icy roads. Your utility function is given by: ug(Z) + ui(D) = In(Z - 3) - In(2 - D). In your ultimate wisdom, you calculate that the cost of driving on the icy roads is 1 unit (So, D=1). What is the minimum value you must obtain from attending the game, so that you decide to go? Solution sent me fast ..Consider a consumer with the utility function U (x1, x2 ) = 10x12/3x21/3 −50. Suppose the prices of x1 and x2 are 10 and 2 respectively and the consumer has an income of 150. (a) Write out the consumer’s constrained optimization problem. Specifically, write out the objective function and constraint for the problem (e.g. max _?_ subject to _?__). (b) Write the Lagrangian equation corresponding to the constrained optimization problem. Derive the Necessary First Order Conditions. (c) Use the NFOCs to solve for the consumer’s optimal bundle. (d) Show that at the solution you found in (c), the tangency condition is satisfied: MRS = p1 / p2. (e) How did the ‘50’ in the utility function influence the optimal con- sumption bundle? How did the ‘10’ in the utility function influence the optimal consumption bundle? (i.e., how would the optimal bun- dle change if these coefficients were to change?). How would the optimal bundle change if the utility function was x12x2? Lastly, how would…