1. ● Three players together can obtain 1 to share, any two players can obtain 0.8, and one player by herself can obtain zero. ● Then, N= 3 and v(1) = v(2) = v(3) = 0, v(1, 2) = v(2, 3) = v(3, 1) = 0.8, v(1, 2, 3) = 1. Which allocation is in the core of this coalitional game? a) (0,0,0); b) (0.4, 0.4, 0); c) (1/3, 1/3, 1/3); d) The core is empty;
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- 23. 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 7 9 2 Room Y 4 3 7 Room Z 2 1 4 Utility for A Utility for B Utility for C (a) How many possible allocations are there in total? (b) Identify the two allocations which are not Pareto optimal and explain why they are not Pareto optimal. (Hint: is the allocation (Y,Z) Pareto timal?) (c) Suppose we square Carlos' utility from each room (i.e. uc (Z) becomes 16). Would the set of Pareto optimal outcomes change? Why/why not? (d) Returning to the utilities from part (a), which of the Pareto optimal allocations maximise total surplus (utility) and would all housemates weakly prefer this allocation over any other? (e) Suppose the housemates decide to allocate…Consider trade between two consumers (1 and 2) and two goods, X and Y. Suppose the total quantities of each good are 100 units. Each consumer has Cobb-Douglas preferences given by: U(X,Y) = XY Denoting by X1 the first consumer's consumption of X and by Y1 the first consumer's consumption of Y, the contract curve consists of all allocations where A. X1 = 0.5Y1 B. X1 = 2Y1 C. X1 = Y1 square D. X1 = Y1Question 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)
- consider an exchange economy with 2 goods (1 and 2) and 2 consumer (A and B). a bundle with x units of good 1 and y units of good 2 is written as (x,y). consumer A has an endowment (4,0) and consumer B has an endowment (12,12). the 2 goods are perfect substitutes for each consumer. consider an allocation in which A receives (1,9) and B receives (15,3) if we can redistribute endowments suitably, it is possible to obtain this allocation as the outcome of a competitive equilibrium. is this true or false? explain carefullyIs the following statement correct: "For any two-person exchange economy, if each consumer has a complete, transitive and convex preference relation, then every competitive equilibrium allocation is efficient"? Provide a proof of your argument. [For this question, maximum word limit is 80.]7 Ann and Bob are dividing a cake made up by 1/3 chocolate parts and 2/3 vanilla parts. Ann likes vanilla twice as much as chocolate and Bob likes vanilla just as much as he likes chocolate. What division of the cake is suggested by the egalitarian solution? By the competitive equilibrium with equal incomes? Also name one allocation that is not Pareto optimal.
- Suppose Hang Tuah has 10 keris and 2 arrows and Hang Jebat has 2 keris and 2 arrows. If this allocation is considered as Pareto Optimal, what mutually beneficial exchange could be made? Andaikan Hang Tuah mempunyai sepuluh keris dan enam anak panah dan Hang Jebat mempunyai 2 keris dan 2 anak panah. Jika peruntukan ini dianggap sebagai Pareto Optimum, apakah pertukaran yang saling menguntungkan yang boleh dilakukan? a. Hang Jebat should give up arrows to get more keris. Hang Jebat patut melepaskan anak panah untuk mendapatkan lebih banyak keris. b. Hang Tuah should give up keris to get more arrows. Hang Tuah patut melepaskan keris untuk mendapatkan lebih banyak anak panah. c. None. No beneficial exchange can result that benefits one party without hurting the other. Tiada. Tiada pertukaran yang berfaedah yang boleh menyebabkan keuntungkan kepada satu pihak tanpa merugikan pihak yang lain. d. Arrows should should be taken from Hang Tuah and given to Hang Jebat to equalize their weapons.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). Assuming3. 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, U1(x1, y1) = 2(x1y1) 4 + 20 U2(x2, y2) = 10lnx2 + 4lny2 W1 = (2, 1) and W2 = (1, 2) 1.1 Characterize the Pareto efficient allocations and contract curve. 1.2 Which allocations are in the core? 1.3 Find the Competitive/Walrasian equilibrium for this problem. 1.4 Is the answer of 1.3 in the core?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.Alice can produce goods X and Y. It takes her 1 hour of labor to produce 2 units of X. It takes 2 hours of labor to produce three units of Y. If Alice divides four hours of her time between the two tasks, which of the following (X, Y) allocations are efficient? Could be more then one answer. (3, 4) (8, 0) (4, 3) (5, 1)