3. Consider an Edgeworth box economy in which the consumers have the Leontief utility func- tions u(11, Y1) by wi = (3, 1) and w = equilibrium prices and allocations. = min{2r1, y1} and u2(12, 42) = min{r2, 2y2}. The endowments are given (1,3). Suppose the prices for both goods are positive. Find the %3D
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- Please 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.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.John 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. J2. Consider an economy with two agents and two commodities. Consumers' preferences are represented by the following utility functions u₁(x,x) = (x²) ¹ (x²) ½ u₂(x², x²) = x² + x². Consumers' initial endowments are e² = (6,4). Note: You can normalize the price of one good to 1 at any point when solving this question. e¹ = (10,2) (a) Draw the Edgeworth box that represents this economy. Clearly indicate the size of the box (i.e. the maximal feasible amounts of good 1 and good 2). Show the location of the initial endowment and draw the indifference curve of each consumer that passes through the initial endowment.Can you help me with this question. Im finding it quite difficult. In a two-good economy there are two equal-sized groups of people: type a have preferences given by 2 log(xa1) + log(xa2) and type b have preferenceslog(xb1)+2log(xb2) where xhi means consumption by a type h person of good i. The division of property is as follows: each a-person has an endowment of (30, k) units of the two goods; each b-person has an endowment of (60, 210−k) units, where k is some given number between 0 and 210. Assume that there is no production and that people can freely exchange goods to maximise their utility. If there is a competitive equilibrium, what are the individuals’ incomes (ya, yb) in equilibrium as a function of k?John 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 J B. b. C < J. c. C = J. d. C = B. e. J5.18 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).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.4. Consider a two-consumer, two-good exchange economy. Utility functions and endowments are u'(x1, x2) = (x1x2)² and e' = (18, 4),u²cx1, x2) = In(x1) + 2 In(x2) and e? = (3, 6). (a) Characterize the set of Pareto-efficient allocations as completely as possible. (b) Characterize the core of this economy. (c) Find a Walrasian equilibrium and compute the WEA. (d) Verify that the WEA you found in part (c) is in the core.Exercise 4 Consider an economy with two consumers, Alexia and Bart, who live two periods, t = 0 and t = 1. In each period they can consume one type of good and their preferences for consumption are given by U (co, c²) = c(c²)² _i = A, B. Alexia and Bart have the following endowment of good in each period M=1, M₁ = 1, MB = 2, MB = 2. In t = 0, Alexia and Bart can exchange a financial contract for the delivery of one unit of consumption good in t = 1 (a bond). Name p the price of the bond and b² the amount bought by agent i = =A, B. (a) Write down each agent's utility maximization and budget constraints assuming that he/she can trade the bond without restrictions. (b) Find each agent's optimal quantity b² as a function of the bond net return r. (c) Find the equilibrium value of r and the equilibrium demand/supply of each agent.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.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. %3DSEE MORE QUESTIONSRecommended textbooks for youPrinciples of Economics (12th Edition)EconomicsISBN:9780134078779Author:Karl E. Case, Ray C. Fair, Sharon E. OsterPublisher:PEARSONEngineering Economy (17th Edition)EconomicsISBN:9780134870069Author:William G. Sullivan, Elin M. Wicks, C. Patrick KoellingPublisher:PEARSONPrinciples of Economics (MindTap Course List)EconomicsISBN:9781305585126Author:N. Gregory MankiwPublisher:Cengage LearningManagerial Economics: A Problem Solving ApproachEconomicsISBN:9781337106665Author:Luke M. Froeb, Brian T. McCann, Michael R. 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