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Suppose X = R k + for some k ≥ 2, and we define x = (x1 , …, xk ) ≥ = (y1 , …, yk ) if x ≥ y; that is, if for each i = 1, …, k, xi ≥ yi . (This is known as the Pareto ordering on R k + ; it plays an important role in the context of social choice theory in Chapter 8.)
(a) Show that ≥ is transitive but not complete.
(b) Characterize ≥ Is asymmetric? Is≥ Negatively transitive? Prove your assertions.
(c) Characterize ~ defined from ≥ in the usual fashion; that is, x ~ y if x ≥ y and y x. Is ~ reflexive? Symmetric? Transitive? Prove your assertions.
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- (a) Examine the view that it is possible to escape from the "impossibility" result of Arrow by making social preferences a function of cardinal utility rather than of individual orderings.3(b)Consider the following social choice problem in the setting of consumption of two goods by two consumers. The two goods are called tillip and quillip and the two consumers are called 1 and 2. Consumer 1 has utility function U; (t,g) =6+.4 In(t) + .6 In(q) (where t is the amount of tillip 1 consumes and q is the amount of quillip). Consumer 2 has utility function U; (t,q) = 8 + In (t) + In(q). The social endowment consists of 15 units of tillip and 20 units of quillip. a. Suppose that a social dictator has social welfare functional of the following form: social welfare, as afunction of (u",u") is a weighted sum with weight 2 on the lesser of u' and u' and weight 1 on the greater of the two. What will be the welfare optimum plan chosen by this social planner? b. What is the set of all feasible, Pareto efficient allocations of the consumption good for this society? %3DIs the pareto principle generally accepted as true or accurate? Or is it considered not true?
- Rafe is optimally choosing to consume 6 apples and 3 bananas. The prices of apples and bananas are p. = 7 and pb - 7. Which of the following utility functions over quantities of apples (a) and bananas (b) could represent Rafe's preferences? = u(a, b)-a4/5 61/5 Ou(a, b) = a¹/5 64/5 Ou(a, b)-a2/3 b1/3 Ou(a, b)-a¹/3 2/3Suppose the economy has 100 units each of goods X and Y and the utility functions of the (only) 2 individuals are: UA (XA,YA) = X0.25Y 0.75, UB (XB,YB) = X0.75Y 0.25 . Show that pareto-improvement is possible if, initially, goods are divided equally between the two individuals.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.
- Consider the following utility functions: a) U= xy b) U=(xy)l3 c) U=min(x,y/2) d) U=2x + 3y e) U=x?y? + xy 1. Check if any of utility functions a) – e) are homogeneous. If so, state the degree of homogeneity of each.There are two shows on Netflix that Jeff can watch, Show 1 and Show 2. x₁ measures the number of episodes watched of Show 1 and x₂ measures the number of episodes watched of Show 2. Jeff's preferences can be represented by u(x₁, x₂) = x² + ax². Jeff has 5 hours available for watching Netflix today. Each episode of Show 1 lasts 30 minutes, while each episode of Show 2 lasts one hour. For all question parts, round to three decimal places if necessary. For parts (a) - (f), let a = 1. Draw Jeff's indifference curves for the utility levels u = 4, 9, 16. Draw a straight line between the bundles (x₁, x₂) = (4,0) and (x₁, x₂) = (0,4). Now, use your picture to determine whether the following two statements are true or false. Answer "1" for true. Answer "0" for false. a) Statement 1: Jeff's preferences are strongly monotone. b) Statement 2: Jeff's preferences are strictly convex. c) Find the bundle at which one of Jeff's indifference curves is tangent to the budget line. (x₁, x₂)=( d) Evaluate…There are two shows on Netflix that Jeff can watch, Show 1 and Show 2. x₁ measures the number of episodes watched of Show 1 and x2 measures the number of episodes watched of Show 2. Jeff's preferences can be represented by u(x₁, x₂) = x² + ax². Jeff has 5 hours available for watching Netflix today. Each episode of Show 1 lasts 30 minutes, while each episode of Show 2 lasts one hour. For all question parts, round to three decimal places if necessary.
- Martha spends 5 hours of free time exercising and playing music per day. The table below represents the level of utility she receives from the various combinations of exercise and music activities. For instance, if she only plays music, her total utility from exercise will be 0, and the utility from music will be 50. If she exercises for 1 hour, she will receive a level of utility of 25. Utility is measured in ‘utils’; those numbers are arbitrary and designed to quantify the benefits received from various actions. Utility for Exercise Utility from Music 0 utils from 0 hours of exercise 50 utils from 5 hours of music 25 utils for 1 hours of exercise 45 utils from 4 hours of music 40 35 45 20 50 0 Marginal analysis will help Martha in her decision making regarding how to spend her free time because ... All answers are correct. She will realize which activity is more important to her. She will be able to find the combination of music and exercise which gives…Martha spends 5 hours of free time exercising and playing music per day. The table below represents the level of utility she receives from the various combinations of exercise and music activities. For instance, if she only plays music, her total utility from exercise will be 0, and the utility from music will be 50. If she exercises for 1 hour, she will receive a level of utility of 25. Utility is measured in 'utils'; those numbers are arbitrary and designed to quantify the benefits received from various actions. Utility Utility from from Exercise Music 50 utils O utils from O hours of exercise from 5 hours of music 25 utils for 1 hours of 45 utils from 4 hours of exercise music 40 35 45 20 50 The utility patterns in the table illustrate the law of marginal diminishing utility because ... The total utility Martha receives from music decreases The increase in utility for each additional hour Martha exercises falls The total utility Martha receives from exercise decreases Martha…Suppose Ed's utility from the consumption of martinis (m) is proportional to the number he drinks: U(m) = m. Further, Ed is particular about his martinis: He enjoys them in exactly three parts of gin (g) to one part vermouth (v). So we can write the utility as: U(m) = U (g, v) = min (g, 3v). (a) Provide a sketch of Ed's indifference curve in the (g, v)-space for various level of utility. (a) Let the prices be p, and p, and Ed's income be I. Find the demand functions for 9 and v [Hint: Use the proportional rule and budget constraint. No need for differential calculus] (b) Using the results in (b), what is Ed's indirect utility function? (c) Find the expenditure function; given a level of utility U, and prices pg and Pv.