3. Consider an Edgeworth Economy with two consumers A and B, and two goods 1 and2. The initial endowments are E₁ = 10 and E2 = 10, and the initial endowments aree₁ = 10, e₁ = 0, e = 0, and e2 = 10. In words, A holds the entire endowment of good1, and B holds the entire endowment of good 2.u(x1, x2)=x1x2,and the utility function of consumer B isuB (y1, y2) = √√√1 + √√√2-(a) Draw the Edgeworth Box that shows the initial endowments on it.(b) Verify that the price ratio ₁ = 1 and x₁ = x² = y₁ = y² = 5 is an equilibrium.P2P2Hint: You do not need to derive the equilibrium from scratch. You can justshow that given the price ratio P₁ = 1, both consumers maximize utility whenconsuming 5 units of each good, and that this allocation is feasible. Alternatively,you can just show that the allocation satisfies the MRSA = MRSB and thefeasibility conditions.(c) Solve for the equilibrium from scratch, instead of guessing and verifying.

The objective of the question is to understand the concept of Edgeworth Economy and to verify the…

Answered: 3. Consider an Edgeworth Economy with…

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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3. Consider an exchange environment with two consumers, Consumer 1 and Consumer 2, and two goods, a and b. Consumer 1 is initially endowed with e₁,a = 0 of good a and e₁,b = 10 of good b, while €1,6 Consumer 2 is initially endowed with e2,a = 10 of good a and €2,6 = 0 of good b. The utility of Consumer 1 from consuming x1, a ER+ of good a and 1,6 ER is u1(x1,a,x1,b) = x1,a + x1,b, while the utility of Consumer 2 from consuming x2,a Є R+ of good a and x2,6 ЄR is U2(x2,a, X2,b) = √√x2,a + x2,b. (a) Consider general equilibrium in the context of this environment in which the price of good b is normalized to 1. Compute the equilibrium price pt of good a and find the corresponding equilibrium quantities consumed by the various consumers. (b) Consider the game in which, simultaneously, Consumer 1 submits a “demand curve" and Consumer 2 submits a “supply curve” to a market mediator. In particular, the strategy space of Consumer 1 is [0,10], and the strategy d of Consumer 1 corresponds to the…
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, ci) = c(ci)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.
2. Assuming Bart and Homer barter (no pun intended), highlight the area of all post-trade allocations both of them prefer to their initial endowment. 3. What determines which particular post-trade allocation they end up with? Marge observes what is going on and intervenes. She (acting as the "Government' in our example) decides that Bart should not have any beer. She makes a lump-sum transfer leading to new initial endowments for Bart (0,21) and Homer (38,0). 4. In your Edgeworth box show the new initial endowments after the lumpsum transfer imposed by Marge. Label this poișt e2. (1 point) 5. After Marge's transfer mark one new possible post-trade allocation (Label this point f.) and draw the associated indifference curves for Bart and Homer. (1 point) 6. Did Marge succeed in her mission to not let Bart have any beer? Why or why not? 7. Add the contract curve to your Edgeworth box. 8. What do all points on the contract curve have in common?
1. Hatfield and McCoy hate each other. The only good they both consume WH W2 40 Mand McCoy's is whiskey. Hatfield's utility function is UH utility function is UM = WM-W, where WHI is Hatfield's consumption 40' Η of whisky and WM is McCoy's consumption of whiskey. The sheriff has confiscated a total of 60 litres of whiskey which he could give back to them or throw in the creek. Say that the Sheriff wants them both to be as well off as possible and weighs their "happiness" equally. How much whiskey should the sheriff give to each of them? Should he throw any whiskey in the creek; if so how much? =
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.
6. 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. (a) The conditions require that A and B consume at a point on their budget lines where their indifference curves have the same marginal rate of substitution. and equilibrium for the market as a whole requires that the sum of the individuals' demands for each commodity must equal the totals (Xo and Yo) available. Mathematically express the point of equilibrium, that is, express the above statement with equations in relating each endowments, equilibrium points (Xa, Xb, Ya, Yb), and totals ((Xo, Yo).
2. Let two consumers have preferences described by the utility function: U =log(x",)+log(x"2), h=1,2 and the endowments described below: Good 1 Good 2 Consumer 1 3 2 Consumer 2 2 3 a. Draw the Edgeworth box for the economy and the initial endowment of consumers 1 and 2. b. Suppose good 1 has a prize p, and good 2 has a price of $1 (called the nummeraire). Write down the budget constraint for both consumers. C. Calculate the consumers' demand functions. d. Find the price of good 1. Then, find the equilibrium levels of consumption.
1. Assume you spend your entire income on two goods X & Y with prices given as PX & PY, respectively. Prices and income (I) are exogenous and positive. Given that U = X2 + Y2 , derive the Marshallian demand function for good Y and evaluate the type of good. 2. Assume you spend your entire income on two goods X & Y with prices given as PX & PY, respectively. Prices and income (I) are exogenous and positive. Given that U= X2Y2 , derive the Hicksian demand function for good Y.3. Suppose that initially PX = 2, PY = 8, I = 96 and the Marshallian demand function for good Y is given by Y∗ = (0.5I/ PY)+(0.5PX/PY)− 0.5. Calculate the own price & income elasticities of demand for good Y. Interpret your computed values and say something about the type of good.4. Suppose the economy has 100 units each of goods X and Y and the utility functions of the (only) 2 individuals are: UA (XA,YA) = X0.25Y0.75, UB (XB,YB) = X0.75Y 0.25Show that pareto-improvement is possible if,…
2) Two consumers R and S share an endowment (x, y) of goods X and Y. The consumers' utilities from consumption can be written as UR = aln(XR) + bln(YR), and Us=aln(xs) + bln(ys), where XR, XS, YR, and ys are the quantities which each consumes, and a and b are parameters, for which a + b =1. Write a report in which you a) Define the concept of the contract curve, and show that for these endowments, it will be the upward diagonal of the Edgeworth box, which shows all possible divisions of the endowment. b) Assume that in the initial endowment, XRE = XE and ysE=yE. R begins with the endowment of X and S begins with the endowment of Y. By setting up a constrained maximisation problem for each consumer, obtain their offer curves, and hence obtain the Walrasian equilibrium. c) State the two welfare theorems, and explain how they apply in this case.
2. There are two consumers A and B and two commodities x and y. ,1-a. UA = x%yAa: Utility function of consumer A. w: consumer A's endowment of good x w: consumer A's endowment of good y UB = 2xB + YR: Utility function of consumer B. w: consumer B's endowment of good x w: consumer B's endowment of good y Px: price of good x Py: price of good y p = Px /py; relative price of good x a. Find the equation of the contract curve. b. Find the competitive equilibrium (p*, x, yÃ, x;, yB). Your answers should be in terms of the parameters.
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.
7. In an Edgeworth box for two consumers, A and B, with endowments of commodities Xa and 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. (a) The conditions require that A and B consume at a point on their budget lines where their indifference curves have the same marginal rate of substitution, and equilibrium for the market as a whole requires that the sum of the individuals' demands for each commodity must equal the totals (Xo and Yo) available. Mathematically express the point of equilibrium, that is, express the above statement with equations in relating each endowments, equilibrium points (Xa, Xb, Ya, Yb), and totals ((Xo, Yo). (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…

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