2(a) Suppose x is a Pareto efficient allocation in which each agent holds a positive amount of each good. that preferences are convex, continuous and monotonic. Then show that x' is a Walrasian equilibrium for the initial endowments that are redistributed such that w, = x, for i = 1,...n.
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- Throughout this problem set, we will look at exchange economies with two goods and two agents. Let X = R², let u denote agent i's utility, and let wie X denote agent i's endowment. 1. Suppose u¹(x¹) = min{ri, 2} and wi = (4,8) for both agents i. (a) Argue that every Pareto optimal allocation has r≥r for both agents i. (b) Argue that every allocation z with r≥r for both agents i is Pareto optimal. (c) Draw an Edgeworth box, with a picture depicting every Pareto-optimal allocation. In this picture, also draw the endowment allocation, and draw each agent's indifference curve through the endowment. (d) Argue that, in any competitive equilibrium, the price of good 2 must be zero. (e) Find all competitive equilibria.Q. It is possible to have a Pareto efficient allocation where everyone is worse off than they are at an allocation that is not Pareto efficient?3. Anita (A), Ben (B) and Carlos (C) are housemates who have moved to a new house and must decide how to allocate rooms X, Y and Z. An 'allocation' is where each housemate is assigned to exactly one room. For example, Anita → Room Z, Ben → Room X and Carlos → Room Y is allocation (Z, X, Y). Utilities for each room are given below: Room X Utility for A 7 Utility for B 9 Utility for C 2 Room Y 4 3 7 Room Z 2 1 4
- Two consumers, Budi and Marry, together have 10 apples and 4 oranges. a. Draw the Edgeworth box that shows the set of feasible allocation for the two individuals and two goods b. Suppose Budi has 5 apples and 1 orange, while Marry has 5 apples and 3 oranges. Identify this allocation in the Edgeworth box c. Suppose Budi and Marry have identical utility functions and assume that this utility function exhibits positive marginal utilities for both apples and oranges and a diminishing marginal rate of substitution of apples and oranges. Could the allocation in part (b) be economically efficient?Consider an economy inhabited by George and Harriet, whose utility functions are Ug : (ac)² (bc)2 Он тан + 2bн The total quantities of ale and bread that can be produced by the economy are a and b, and they are constrained by the production function b = 2(10 – a)/2 There are infinitely many Pareto optimal allocations. In one of them, Harriet's utility is 8. a) An allocation in this economy is described by a list of four variables. What are these variables? b) What four equations describe the Pareto optimal allocation in which Harriet's utility is 8? c) Find this Pareto optimal allocation.True/False (1) If the assumptions of the first theorem of welfare economics apply and if the economy is in a competitive equilibrium, then any reallocation that benefits someone must harm someone else.< (2) The second welfare theorem of economics states that if preferences are convex, then any Pareto optimal allocation could be achieved as a competitive equilibrium after some reallocation of initial endowments.<
- Suppose Zaharah (Z) and Quah (Q) are now at an allocation bundle where their marginal rates of substitution (MRS) are given by MRS₂ = 10 and MRSQ = 5, respectively. In general equilibrium analysis, this suggests that Andaikan Zaharah (Z) dan Quah (Q) kini berada pada sesuatu longgok alokasi di mana kadar penggantian marginal (MRS) masing-masing adalah MRS₂ = 10 dan MRSQ = 5. Dalam analisis keseimbangan am, ini menunjukkan bahawa a. both individuals have already engaged in voluntary exchange of resources kedua-dua individu telah pun terlibat dalam pertukaran sumber secara sukarela b. Zaharah's situation is better off than Quah situasi Zaharah adalah lebih baik daripada Quah there exists a basis for voluntary exchange for resources between both terdapat asas untuk pertukaran sumber secara sukarela antara kedua-duanya d. both of them are in equilibrium C.Q.2 (a) (i)Define "weakly Pareto efficient" and "strongly Pareto efficient" allocations. (ii) show that an allocation that is 'strongly Pareto efficient' is also 'weakly Pareto efficient. But in general the reverse is not true. (ii) If preferences are continuous and monotonic prove that an allocation is 'weakly Pareto efficient' if and only if it is 'strongly Pareto efficient'.9. Consider an Edgeworth box economy with two consumers, whose utility func- tions and endowments are e' = (5,5) 2 = (5,5) In the following, use the normalization p2 = 1. (a) Find the competitive equilibrium price. (b) State the first fundamental theorem of welfare and verify that it holds in this economy. (e) Consider the allocation ã = (x',) = (2,3), (8, 7). Show whether this allo- cation can supported as an equilibrium with transfers. (d) State the second fundamental theorem of welfare, and briefly discuss whether the result in part (c) conform with or violate this theorem.
- Consider an economy with 100 pieces of apple (A) and 150 pieces of banana (B) that must be completely distributed to individuals 1 and 2. The utility function of the two individuals, U, & U2, is given by U1 (A1,B1) = 2A? + B² & U2 (A2,B2) = 2A²B?, respectively. With this information, recommend an efficient allocation of the two %3D goods between the two individuals. Discuss and show the necessary solution to support your recommendationJohn and Belle consume only two goods, x and y. They have strictly convex preferences and no kinks in their indifference curves. At the initial endowment point, the ratio of John's marginal utility of x to his marginal utility of y is J and the ratio of Belle's marginal utility of x to her marginal utility of y is B, where ] B. b. C < J. c. C = J. d. C = B. e. J16.11. Ted and Joe each consume peaches, x, and plums, y. The consumers have identical 10y7x7, MRS = 10yr^TTogether, they have 10 peaches MRSJoe utility functions, with and 10 plums. Verify whether each of the following allocations is on the contract curve: a) Ted: 8 plums and 9 peaches; Joe: 2 plums and 1 peach. b) Ted: 1 plum and 1 peach; Joe: 9 plums and 9 peaches. %3DSEE MORE QUESTIONS