A social welfare function is thought to be additive along the lines proposed by Bentham and Mill: + Un W = U, + U2 + ... + where U; is individual i's utility and there are n individuals. Suppose an economy is made up of two individuals, John and Jane. John's utility function is Ujohn = 28/X and Jane's utility function is UJane = 2X. %D There is a fixed quantity of good X (100 units) evenly divided between John and Jane (50 units for each). Social welfare (W) is equal to: W = utils. (Enter your response rounded to the nearest whole number.) Social welfare could be improved if a portion of good X were transferred from to John Jane

A social welfare function is thought to be additive along the lines proposed by Bentham and Mill: + Un W = U, + U2 + ... + where U; is individual i's utility and there are n individuals. Suppose an economy is made up of two individuals, John and Jane. John's utility function is Ujohn = 28/X and Jane's utility function is UJane = 2X. %D There is a fixed quantity of good X (100 units) evenly divided between John and Jane (50 units for each). Social welfare (W) is equal to: W = utils. (Enter your response rounded to the nearest whole number.) Social welfare could be improved if a portion of good X were transferred from to John Jane

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter4: Utility Maximization And Choice

Section: Chapter Questions

Problem 4.11P

See similar textbooks

Similar questions

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.
Person 1 and person 2 are the only two residents of an economy. Person i (where i is either 1 or 2) has the utility function Here, Yi is person i’s income and β is parameter between 0 and 1. Assume that the social welfare function is where α is a parameter between 0 and 1. Initially, person 1’s income is 1 and person 2’s income is 2. a) Express W in terms of Y1 and Y2. If a social indifference curve shows all the pairs (Y1, Y2) that yield the same value of W, what would a social indifference curve look like if it were drawn in the (Y1, Y2) quadrant? Find an algebraic expression for the slope of a social indifference curve. b) Imagine that income redistribution is costless, in the sense that the economy can reach any pair (Y1, Y2) that satisfies the condition Draw a graph of the attainable pairs in the (Y1, Y2) quadrant. This set is called the “utility possibility frontier.” Using this frontier and the social indifference curves,…
The utility possibility frontier for two individuals is given byUA + 2UB = 200.(a) On the graph paper provided, plot the utility possibility frontier.(b) Based on the function W(UA, UB) =max{UA, UB} on the utilitypossibility frontier in part (a), in order to maximize a Nietzschean social welfarefunction, what should be the values of UA and UB?(c) If we use a Rawlsian criterion, W(UA, UB) =min{UA, UB}, in orderto maximize the social welfare function, what should be the values of UA and UB?(d) If the social welfare is given by W(UA, UB) = U1/2A, U1/2B, in thiscase, with the utility possibility frontier in part (a) what should be the values ofUA and UB if social welfare is to be maximize?On the same graph plotted in part (a), show the three socialmaxima by drawing a Nietzschean isowelfare line through the Nietzscheanmaximum, a Rawlsian isowelfare line through the Rawlsian maximum and aCobb-Douglas isowelfare line through the Cobb-Douglas maximum.
If the social welfare function produces concave social welfare indifference curves, social welfare can be increased by reallocating income between households if their marginal utilities of income differ. TRUE OR FALSE
Consider an economy composed of 16 consumers. Of these, 5 consumers each own one right shoe and 11 consumers each own one left shoe. Shoes are indivisible. Everyone has the same utility function, which is Min(2R, L}, where R and L are, respectively, the quantities of right and left shoes con sumed. A) (10%) Is the status quo (where each individual has his own shoe) Pareto efficient? If so, briefly explain why. If not, provide a Pareto improvement b) (10%) Characterize all Pareto efficient allocations
Is Pareto Efficiency enough to explain Welfare economics?
Consider an Edgeworth box economy, where preferences and endowments are given by. ( attached below ) In the following, use the normalization p2 = 1.(a) Find all the Pareto optimal allocations (use x1 to parameterize thePO allocations). (b) Suppose the government wants individual 1 to have the bundle x1 = (5, 9). Show whether the government can induce this outcome an equilib- rium with transfers. Find the supporting prices and transfers if possible. (c) State the Second Fundamental Theorem of Welfare and explain how the result in part (b) relates to that theorem. (d) Suppose the government does not care what bundle individual 1 gets and only wants to ensure that the individual 1 receives utility level equal to 45. Show whether this can be achieved as an equilibrium with transfers. Find the supporting prices and transfers if possible.
Because Income depends on work effort and work effort depends on nutrition, suppose the relationship between income (Y) and total food availability (F) in a poor household is given by: Y = 10F + 20F2 – 0.6F3 a) If food availability in a two-person household is 20, would it be “rational” to share the food? Explain. b) If food availability in a two-person household is 30, would it be “rational” to share the food? Explain.
The most general form of a social welfare function (SWF ) can be written as W = W(U1; ... ; UH). a. Explain the following properties that a SWF may satisfy: non paternalism, Pareto principle, anonymity (the names of the agents do not matter), and concavity (aversion to inequality). b. Consider two agents h = 1; 2 with utilities U1 and U2. Depict the social indifference curve of the utilitarian SWF in (U1; U2)-space. Which of the properties in part a does it satisfy? c. Depict the social indifference curves of the maxim in or Rawlins SWF. Contrast to the utilitarian SWF with respect to the aversion to inequality. Which properties does the Rawlins SWF satisfy? d. The Bernoulli-Nash social welfare function is given by the product of individual utilities. Discuss the distributional properties of the Bernoulli-Nash SWF.
Consider the utility functions below of two individuals, A and B, and bundles of goods Q and R. UA=X0.5Y0.5; UB=X+2Y; Bundle Q (10, 10); Bundle R (10, 15). Suppose the total X and total Y available in the economy are both equal to 20. a. If initially both individuals are consuming bundle Q, then a pareto-improvement is possible through reallocation of goods, i.e. individual A gives B some of his good X in exchange for some of individual B’s good Y. (True or False? Explain through mathematical examples).b. Pareto-optimality is achieved if we give individual B Bundle R and the remaining goods X and Y available in the economy is given to individual A. (True or false? Explain through a graphical example)
Explain the concept of the utility possibility frontier. What is the role of the utilitypossibility frontier in the social welfare maximization problem of a welfarist society?
Consider a simple economy with two individuals (A and B) and two goods (x and y). Can you please Write down the Pareto efficient conditions for this economy and explain thefirst fundamental theorem of welfare economics?