First Welfare Theorem: Suppose we have an endowment economy with two consumers i=1,2. Let e = (1, 1) be the endowment for both consumers. Let preferences be given by U¹(x¹, y¹; x²) := (x¹y¹)¹/² + 7x² and U²(x², y²; y¹) := (x²y²)¹/2 + 7y¹. When maximimizing utilty each consumer takes the others choices as given. Show that p = (1,1) is a competitive equilibrium where each consumer demands her own endowment. Find a feasible allocation that is a Pareto improvement. Why does the first welfare theorem not apply? А

First Welfare Theorem: Suppose we have an endowment economy with two consumers i=1,2. Let e = (1, 1) be the endowment for both consumers. Let preferences be given by U¹(x¹, y¹; x²) := (x¹y¹)¹/² + 7x² and U²(x², y²; y¹) := (x²y²)¹/2 + 7y¹. When maximimizing utilty each consumer takes the others choices as given. Show that p = (1,1) is a competitive equilibrium where each consumer demands her own endowment. Find a feasible allocation that is a Pareto improvement. Why does the first welfare theorem not apply? А

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter4: Utility Maximization And Choice

Section: Chapter Questions

Problem 4.11P

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4. Aaron and Burris have the following utility functions over two goods, x and y. Aaron’s utility function: UA(xA, yA) = min{xA/3, yA} Burris’s utility function: UB(xB, yB) = 9xB + 3yB Aaron’s endowment is eA = (2, 4). Burris’ endowment is eB = (10, 8). In an Edgeworth Box diagram, show which allocations are in the core. Solve for the set of Pareto optimal allocations (i.e. the contract curve) in the Edgeworth Box. Illustrate the contract curve in an Edgeworth Box diagram. Let good y be the numeraire (i.e. set py = 1 and let px = p). Solve for the Walrasian competitive equilibrium allocation and price ratio.
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.
2. 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.
Graph make graph
Suppose there are two consumers, A and B. There are two goods, X and Y. There is a TOTAL of 8 units of X and a TOTAL of 8 units of Y. The consumers' utility functions are given by: UA(X,Y) = 2X + Y UB(X,Y) = X*Y2 Which of the following allocations is Pareto Efficient? None of the other answers are Pareto Efficient. Consumer A gets 3 units of X and 8 units of Y, and Consumer B gets 5 units of X and O units of Y. Consumer A gets 4 units of X and 4 units of Y, and Consumer B gets 4 units of X and 4 units of Y. Consumer A gets 1 units of X and 4 units of Y, and Consumer B gets 7 units of X and 4 units of Y. Consumer A gets 8 units of X and 8 units of Y, and Consumer B gets 0 units of X and O units of Y.
Refer to figure. Suppose the consumer is endowed with 10 units of orange and consumes 5 units of apple. The price of the apple decreases and at the new price the consumer consumes 9 units of apple. The change in the demand for apples due to the endowment effect is equal to_____
First Welfare Theorem: Suppose we have an endowment economy with two consumers i = 1,2. Let ei given by (1, 1) be the endowment for both consumers. Let preferences be U (r', y'; x²) := (x*y*)'/2 + 7x² and Ư°(x², y²; y') := (x²y?)'/2 + 7y'. When maximimizing utilty each consumer takes the others choices as given. Show that p* = (1,1) is a competitive equilibrium where each consumer demands her own endowment. Find a feasible allocation that is a Pareto improvement. Why does the first welfare theorem not apply?
Refer to figure. Suppose the consumer is endowed with 10 units of orange and consumes 5 units of apple. The price of the apple decreases and at the new price the consumer consumes 9 units of apple. The change in the demand for apples due to the endowment effect is equal to Optionsa) 3b) 4c) 1d) none of these
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. J
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. J
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.
Question 1. (i) please!