(a) Consider an agent whose preferences over any couple (1, 2), where x₁ € R+ and 2 € R+, e.g., apples and oranges, is such that she prefers the bundle that is closest to having the same number of apples and oranges. Write a utility function u: R2 → R+ which represent these preferences.
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- 2 Utility representations Recall that for a finite outome set X, the rank-score function U# (x) = #({x' = X : x ≥ x'}) is a utility function representation for any preference relation. In this problem we'll consider the case of a countably infinite outcome set X = {x¹, x², x³, ...}. 1. Write down a preference relation over X for which the rank-score function is a utility representation. Write down another preference relation for which it is not a utility representation. 2. Consider the preference relation over X defined by x¹ ≥ x¹ iff i ≤ j. Write down a utility representation which assigns non-positive utility to every outcome. Write down another utility representation which assigns non-negative utility to every outcome.5. Sheila and Bruce are taking a canoe trip. Sheila brought 10 boxes of peanuts (x) and 15 bags of chips (y). Sheila's utility function is U*(x,y) = lnvx*+ Invy°. Bruce also brought 20 boxes of peanuts and 5 bags of chips. Bruce's utility function is UB(x.y) = min[x', y'I. a) Illustrate the endowment point and draw a sample set of indifference curves through the endowment point. b) If Sheila and Bruce trade what will be the pattern of mutually beneficial trade? c) If the terms of trade are the number of bags of chips (y) per box of peanuts (x) then what is the largest value that these terms can be for a mutually beneficial trade in this economy? d) Find one mutually beneficial trade where the terms of trade are 1 bag of chips (y) per 2 boxes of peanuts (x). Suppose that Sheila and Bruce set up two competitive markets for peanuts and chips. Below you will show that if the price of peanuts (x) is $1 and the price of chips (y) is $2 then the markets for both peanuts and chips will clear.…6. Sheila and Bruce are havinga tea party. Sheila is bringing 12 ounces of tea (x) and 15 finger sandwiches (y). Bruce isbringing 12 ounces of tea and 12 finger sandwiches. Sheila's and Bruce's preferences over tea and sandwiches can be represented by the utility functions: U*(x°,y*) = 6lm x³ + 17la y and UP(x".y") = la x³ + lm yB. a) What is Sheila's MRS at her endowment? What is Bruce's MRS at his endowment? Is the endowment point Pareto efficient? Illustrate in an Edgeworth box diagram the endowment point and a pair of indifference curves through the endowment point. Instead of trading directly with each other Sheila and Bruce trade with a market. Sheila and Bruce are both price takers on both the tea and sandwich markets. Each can buy and sell a good at the same price. Let Px be the price of an ounce of tea and Py be the price of a sandwich. b) Write the two equations that define the best bundle for Sheila given that the prices are Px and Py. Do the same for Bruce. Consider the…
- 2. Consider a consumer whose preference relation over the consumption set X = R can be repre- sented by the utility function: u (x1, x2, 13) = A (x1 – T1)ª (x2 – ã2)° (x3 – ã3)" - where A, a, B and y are all strictly positive. (a) Why can you assume A = 1 and a + B+y= 1 without loss of generality? Do so for the rest of the question. (b) Solve the UMP and derive the consumer's Walrasian demand. (c) Find the indirect utility function, the minimum expenditure function, and the Hicksian demand function.Suppose Jimi has reference dependent preferences over guitars and money as in Tversky and Kahneman (1991). His utility functions are given below. Gains Gains 400 -2 -2 2 Guitars 2$ Losses Losses i-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., "10o0", not "$1000"If Utility=.5lnX+3Z+10+4VW then Marginal utility from X is...... and the Mu of Z is...... Select an answer and submit. For keyboard navigation, use the up/down arrow keys to select an answer. a b с 10; 3 3,4 5/X; 3 d 2/VW; 3
- 4.If we observe a consumer choosing (x1, x2) when (y1, y2) is available one time, are we justified in concluding that (x1, x2) > (y1, y2)?For problems 1, 2, 3, we consider an agent Ann who consumes goods and y and has a utility U (x, y) = x²y. In these problems, Good costs P and good y costs Py. (1) Ann has I to spend. (a) Write out Ann's utility maximization problem. (b) Find: MU, MUy, and MRSxy. (c) Write out Ann's tangency condition. What is the other equation needed to solve for Ann's demand? (d) Find Ann's demands for x and y as functions of Pr, Py, and I. (Partial credit if you do this only for I = 30, P₂ = 5 and Py = 2.)Praxilla, who lived in ancient Greece, derivesutility from reading poems and from eating cucumbers.Praxilla gets 30 units of marginal utility from her firstpoem, 27 units of marginal utility from her secondpoem, 24 units of marginal utility from her third poem,and so on, with marginal utility declining by three unitsfor each additional poem. Praxilla gets six units ofmarginal utility for each of her first three cucumbersconsumed, five units of marginal utility for each of hernext three cucumbers consumed, four units of marginalutility for each of the following three cucumbersconsumed, and so on, with marginal utility declining byone for every three cucumbers consumed. A poem coststhree bronze coins but a cucumber costs only one bronzecoin. Praxilla has 18 bronze coins. Sketch Praxilla’sbudget set between poems and cucumbers, placingpoems on the vertical axis and cucumbers on thehorizontal axis. Start off with the choice of zero poemsand 18 cucumbers, and calculate the changes in…
- Question 1 There are four utility functions defined as follows: U(r, y) 5 ln r +6 ln y U2(x, y) = r+y U3(x, y) r'y U(a, y) = r/y +9 1. Show that the utility functions U1(), Us() and Us() represent the same preference ordering. 2. Show that utility functions U2() and Ua(-) do not represent the same preference ordering. 3. Draw the indifference curve of each utility function at threshold 10 and shade in the upper contour set. Based on your plots, state whether each utility function is quasi-concave or not.You are maxiimizing utility when :A)(MU of X)/(P of X) < (MU of Y)/(P of Y). B)(MU of X)/(P of X) > (MU of Y)/(P of Y). C)(MU of X)/(P of X) = (MU of Y)/(P of Y). D)(MU of X)/(P of Y) = (MU of Y)/(P of X).I am unsure the direction the utility functions would go in , with this specific scenario