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A fixed amount
of a good has to be allocated between two individuals, h = 1; 2 with utility functions
Where xh is the amount of the good allocated to consumer h.
a. How should be allocated to maximize a utilitarian SWF? Illustrate the answer graphically. How do the optimal values of x1 and x2 change among the cases a1 2, a1 = a2, and a1 > a2?
b. What is the allocation maximizing the Bernoulli-Nash SWF ? Illustrate graphically. How do the optimal values of x1 and x2 change with the preference parameters a1 and a2?
c. What is the allocation maximizing the maxim in-Rawlins SWF? Illustrate graphically. How does the allocation change with preference parameters a1 and a2?
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- Let there be 3 people in the economy. Let the utility function of person 1 be u=min(x,y), utility of person 2 be = max(x,y) and the utility function of the 3rd consumer be U=x+y. Let the endowment points be A(5,5), B(5,5) c(5,5). An example of a pareto efficient allocation is: A) A(5,5), B(5,5) c(5,5) B) A(5,5), B(5,0) c(10,5) C) A(5,5), B(0,5), c(5,10) D) None of the above The correct answer is D. Explain clearly. Also, please state the pareto effient outcome not mentioned in the options (if it exists).Considering the given possible uses of candy, which of the four policies could be Pareto Efficient (assuming candy is a “good”)?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.
- Consider a pure exchange economy, where each consumer has preferences described by a Cobb-Douglas utility function. Both consumers have exactly the same endowment. In such an economy, an equitable distribution of goods (where each individual consumes exactly half of each good) is a Walrasian equilibrium allocation? a. always, for any consumers' preferences b. only if consumers' preferences are exactly the same c. Never d. None of the aboveSuppose that 2 roommates, Andy and Bob, are trying to pick an apartment in Chicago. Locations can be chosen from set of alternatives A={ x: x exists [0,1]}. Andy and Bob both want to minimize their daily commute but they work at different locations: Andy at xA=0.3, while Bob at xB=0.6. Specifically, their utility functions are: ui(x)= -(x-xi)2. Question: What is the set of all Pareto Efficient outcomes in A, assuming no money can be exchanged.Assume your apartment complex decides to build new open access common area for its residents. Each individual has value for using the common area represented by the following value function: V= 40-0.5Q where Q represents the number of individuals using the common area. The marginal value function is: MV= 40-Q Assume each individual’s opportunity cost of time is $8 [e.g. MC=8] A) How many people use the common are at the social optimum? B) Calculate the net value (NV) at this social optimum. C) What characteristic of the good must be changed in order to avoid over use of the common area?
- 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.Considering the given possible uses of candy, which of the four choices could be Pareto Efficient (assuming candy is a “good”)? Among the four choices, which policy would yield the highest value according to a “Rawlsian” welfare function? (assuming that sweet/candy is a “good”) Explain. CHOICE 1: Every student gets one candy. The remaining candies are thrown in the bin. CHOICE 2: One student gets all the candies CHOICE 3: Every student gets two candies except for one student, who gets no candy; the remaining go in the bin. CHOICE 4: All the candies go in the binHow Bergson-Samuelson Social Welfare Function used in a model that considers a society's desire for both Pareto optimality and fairness, Mention all necessary assumptions and specify the conditions that ensure a general equilibrium and social welfare
- Two individuals, Fred and Helen, are in an economy with no production, and each have the utility function U = 10XY. Prices of both X and Y are set at $1. Initial endowments for Fred are 10 units of X and 6 units of Y. Helen has 8 units of X and 12 units of Y. Find the general equilibrium prices and allocation, then show that the G.E. allocation is Pareto efficient.The U.S. imports lumber among other goods from Canada. The lumber has been the topic of political dialogue between the 2 countries in the past few days. So let’s take X as ‘lumber’ and Y as ‘other goods’. Suppose the Canadian lumber and other exportable goods are produced with the following production Possibilities frontier (PPF): X2 + Y2 = 900 Suppose the U.S. utility function for the Canadian lumber, X, and other exportable goods, Y, is: U = X Y a. Derive the conditions necessary to determine the ‘Optimal Product Mix’ discussed in chapter 10. b. Draw the PPF (partial just the first quadrant) with Y on the vertical axis. On the same graph, draw at least one indifference curve so as to show the “optimal product mix’. Can this graph represent the ‘pure exchange’ economy? Explain why or why not. c. Now draw in a price line that can make the ‘optimal product mix’ a competitive exchange equilibrium. Can…is the provided Edgeworth box a good example of second-welfare economics theorem.