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- 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). Assuming5. Devin and Giannis consume books and videos. Devin has the endowment (30, 8) while Giannis has (50, 12). Each has the utility function u(b, v) = 4vb + 4vv. (i) Sketch the Edgeworth Box. (ii) Find the equation for the contract curve. (iii) If pv = 1, what is the equilibrium value of pB? (iv) Now suppose that Devin and Giannis both have the utility function u(b, v) = min{½b, v} instead. If pv = 1, what is the equilibrium value of pB?Q. Consider two rational behaving consumers, A and B, in a two-good exchange economy. Their utility functions are defined as follows: 1A 2A X1/2X¹/3 X1/3 X2B 1B Their initial endowments are given by w₁ = (8,5) and wB = (4,3). a. Describe the initial condition that will lead to an exchange. After the exchange, how many units of Good 2 will Individual B end up receiving/offering in the final allocation? Elaborate in detail on the steps towards the solution and round up the final answer to two decimal places. UA UB - = b. Sketch an Edgeworth Box precisely showing the initial allocation and the final allocation on the vertical axis. You do not have to sketch the budget constraint and the indifference curves.
- Suppose that 2 agents, A and B, have preferences over goods 1 and 2: u+(x4) 2.xf + x and u (x²) = xfx. Total endowments are ej = e2 = 4, (a) Find the contract curve. (b) If initial endowments are e4 (4, 0) and eB (0, 4), what is the core?d) An exchange economy has two goods (apples, bananas) and two types of agents (1, 2). Endowment of agent 1 is (3 bananas, 3 apples), and endowment of agent 2 is (4 bananas, 2 apples). Both goods are divisible goods, so that it is possible to consume frac- tions of each good (e.g. 4.92 bananas and apples). Each type 1 agent's preferences are represented by the utility function U(x,x) = min{x, x}, where x and x denote the agent's consumption of bananas and apples, respectively. Each type 2 agent's preferences are represented by the utility function U(x3, x²) = 7min{x3, x4} where x and r² denote the agent's consumption of bananas and apples, respectively. Use an Edgeworth's Box to depict the set of Pareto efficient allocations in this economy. Please capture consumption of bananas on the horizontal axis.5.18 In a two-good, two-consumer economy, utility functions are u¹ (x₁, x₂) = x₁(x₂)², u² (x₁, x₂) = (x₁)²x₂. Total endowments are (10, 20). (a) A social planner wants to allocate goods to maximise consumer 1's utility while holding con- sumer 2's utility at u² = 8000/27. Find the assignment of goods to consumers that solves the planner's problem and show that the solution is Pareto efficient. (b) Suppose, instead, that the planner just divides the endowments so that e¹ = (10, 0) and e² = (0, 20) and then lets the consumers transact through perfectly competitive markets. Find the Walrasian equilibrium and show that the WEAs are the same as the solution in part (a).
- 4) preferences represented by U, (x11, x21) = x11X21. Agent 2 has preferences represented by utility function U2(x12, xX22) = X12X22. The initial endowments are wi = (1,3) and w2 = (3,1). Show directly that every interior Pareto optimal allocation in this economy is a price equilibrium with transfers by finding the associated prices and transfers. Consider a two agent and two good exchange economy where agent 1 has6. In an Edgeworth box for two consumers, A and B, with endowments of commodities Xa and d Ya are A's endowments, Xb and Yb are B's, and Xo = Xa + Xb, Yo = Ya + Yb), the competitive equilibrium allocation of the two commodities represents a mutual tangency of both consumers' indifference curves with each other and with a common budget line. (b) Suppose MRSa = Ya/Xa, MRSb = Yb/Xb, Xa = 10, Ya, = 100, Xb= 50, Yb = 20, and let good Y be the numeraire (Py = 1). Verify that the competitive equilibrium price is Px = 2.Consider a two-good exchange economy with two types of consumers. Type A have the utility function And an endowment of 3 units of good 1 and k units of good 2. Type B has the utility function And an endowment of 6 units of good 1 and 21 - k units of good 2. a. Find the competitive equilibrium outcome and show that the equilibrium price p* = p1/p2 of good 1 in terms of good 2 is p* = 21+k/15. b. Find the income levels (MA; MB) of both types in equilibrium as a function of k. c. Suppose that the government can make a lump-sum transfer of good 2, but it is impossible to transfer good 1. Use your answer to part b to describe the set of income distributions attainable through such transfers. Draw this in a diagram. d. Suppose that the government can affect the initial distribution of resources by varying k. Find the optimal distribution of income if (i) the SWF is W = log(MA) + log(MB) and (ii) W = MA + MB.
- 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.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 %DPlease answer every part. 4. Consider an economy consisting of two individuals, Ann and Bob, and two goods, scotch and wine. Aun has 5 bottles of scoteh and 2 bottles of wine as her endowment, while Bob has 3 bottles of each. Suppose their preferences are described by the following utility functions uA(s, w) = sw and up(s, w) = s'u. Assume also that the prices of goods scotch and wine are represented by P,= 1 (scotch is the mumeraire), and P>0. a. Sketch the Edgeworth box of the economy with Ann at the lower left corner and Bob at the upper right corner; scotch on the horizontal axis, and wine on the vertical axis. Indicate the endowment point e in the box. b. Write the budget lines for Ann and Bob. e. Solve Ann's utility maximization problem. Expross Ann's optimal consumption bundle in terms of P. d. Solve Bob's utility maximization problem. Express Bob's optimal consumption bundle in terms of P. e. Define competitive equilibrium. Compute and plot the CE for this problem.