Consider a two-good exchange economy withr identical consumers. The consumption set is R, the individual endowments are e R? ,, and the preferences are continuous and strongly monotone but not necessarily convex. Argue that the symmetric allocation in which + + every consumer gets her initial allocation is either a Walrasian equilibrium (for some price vector p) or, if it is not, then for r large enough it is not a Pareto optimum. [Hint: the differentiable case is simpler.]
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- Consider a two-person endowment economy characterized by the Edgeworth Box below: a. b. C. X2. OA A E B OB X1 Are the allocations depicted at points A and B Pareto efficient? Explain your reasoning. Add bowed indifference curves that reflect the utility that the two consumers get from the endowment depicted at point E (assume that they value both goods). What is meant by the term "Pareto improvement." In your diagram highlight the set of Pareto improving allocations given the endowment.16.11. 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. %3DQuestion 6: Transitivity There is only one consumption good in the world (cities). Karen is moving – suppose that her preferencesover cities are described as follows: she strictly prefers city A to city B if either (1) the average rent in cityA is cheaper than in city B, or (2) the average rent is the same in both cities, and there are more goodrestaurants in city A than in city B. Are these preferences transitive? Justify your answer.
- 2. Consider a two person pure exchange economy with two divisible goods: : a consumer can consume any positive amount of any good The goods are; x1 and x2. The utility function are u' (x1, x2) = x1+Vx2, and u?(x1, x2) = x1 + x2, and the initial endowments are el Pi = 1, compute the competitive equilibrium for this economy. It is to say that you need to find the vector of prices, and allocations that sustain the Walrasian equilibrium. (25, 75) and e? = (75, 25). AssumingZach's preferences are representable by the utility function u = 90.3927, where 9₁ and 92 denote his consumption of goods 1 and 2. (Answers to each of these questions are rounded, where required, to two decimal places.) Still assuming endowments of e₁ = 5 and e₂ = 9 and market prices p₁ = 20 and p2 = 30, what is the maximised value of Zach's utility? O Au= 6.74 O B. u = 5.39 O Cu = 5.65 O D.u = 7.56 O E. None of the aboveAssume the setting outlined in Question 1. Suppose that the social planner considers it to be imperative that agent B consumes exactly one unit of good 1l and four units of good 2. Although the social planner can not force the individuals to a particular consumption, they can enforce transfers of good 1 between the consumers (transfers of good 2 are not enforceable by the social planner). What transfer of good 1 would guarantee that in the resulting competitive Walrasian equilibrium consumer B consumes one unit of good 1 and four units of good 2? Select one: O a. One half unit of good 1 has to be transferred from agent A to agent B. O b. There is no endowment for which agent B would consume x = in the corresponding competitive equilibrium. Therefore, no such transfers exist. 1 and æ = 4 O c. One unit of good 1 has to be transferred from agent A to agent B. O d. Transferring good 1 is not sufficient. It is necessary to transfer one unit of good 2 from agent A to agent B.
- Consider a society consisting of just a farmer and a tailor. The farmer has 30 units of food but no clothing. The tailor has 60 units of clothing but no food. Suppose each has the utility function U=Fc, If the price of clothing is always $1, and the food price is currently $1, then we can conclude O the market is at a competitive equilibrium. the price of food will drop towards a contitive equilibrium. the price of food will increase towards a competitive equilibrium. O None of the above..Sarah and Andrew are two traders in a pure exchange economic with two goods, Bikes (B) and Computers (C). Sarah's preferences are described by the Cobb-Douglas Utility function: U, = B!³ C?3 1/3 S. Andrew's preferences are given by: UA = B}{²C}2 ´A Assume the price of Bikes is 1 and the price of computers is p. The initial endowments are BA = 10, Bs = 20, CA = 20 and Cs= 10. What is the equilibrium price of computers relative to bikes (p)? %3D %D4. a. Consider a consumer with preferences defined over x and y. Demonstrate that it is possible theywould choose to consume some of both commodities when their income is I but would choose toconsume only x when their income is I’ > I. (Remember: if you can draw it without violating anyof the basic assumptions on preferences, it could happen.) b. Conversely, demonstrate that it is possible they would choose to consume only x when theirincome is I but would choose to consume x and y when their income is I’ > I.c. Finally, show it is possible that they would choose to consume only y when their income is I butwould choose to consume only x when their income is I’ > I.
- 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. Consider a consumer with preferences defined over x and y. Demonstrate that it is possible theywould choose to consume some of both commodities when their income is I but would choose toconsume only x when their income is I’ > I. (Remember: if you can draw it without violating anyof the basic assumptions on preferences, it could happen.)Question 29 Assume in the market with two consumers, Chloe and Oliver. Chloe's utility function is U(x, y) = 2x + 120y¹/2 Oliver's utility function is U(x, y) = 2x+10y. At a Pareto optimal allocation in which both individuals consume some of each good, how much y does Chloe consume? I O 60 05 O 225 O 36 Question 30 Consider the game below where the first payoff belongs to player 1 and the second payoff belongs to player 2: (1,3) (2,1) (1,0) (0,2) B D F H 2 П 2 (1,1) A C E G Use backward induction, a Nash equilibrium outcome of the game is: O (1,3) O (2, 1) O (1,0) O (1, 1)