Consider an economy with two consumers A and B and two goods 1 and 2. The utilities of the consumers are given by: uд(x, x) = x^x^ and u¸ (x, x²) = xỉ x² Their initial endowments are (w₁, w^) = (30,300) and (w₁, w₁) = (170, 100) If the two consumers exchange goods with each other in the course of trade, what is their equilibrium allocation? a. (x, x) = (150, 150), (x, x) = (50, 250) O b. None of the other responses is correct Oc. (x, x) = (75, 125), (x, x) = (125,275) Od. (x, x)=(30, 300), (x,x) = (170, 100) Oe. (x,x) (100, 200), (x, x) = (100,200) Of. (x, x) = (50, 200), (x, x) = (150,200) Og. (x, x)=(60, 100), (x, x) = (140, 300) Oh. (x,x)=(150, 180), (x,x) = (50,220) O i. (x, x) = = (90, 180), (x, x) = (110,220)
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- A small exchange economy is comprised of two of individuals, A and B, and two types of goods, x, ,x,. The individuals' preferences over two goods are can be represented by the following utility functions: U*(x;,.x; ) = min (2.x,x; ) and U* (x;,x;) = min(x;.2x, ). Initial endowments are 10 x, (individual A), and 10 x, (individual B). Calculate the price ratio which yields an equilibrium in the exchange market.4) Consider a pure exchange economy with two goods, (x, y), and two consumers, (1, 2). Consumers' endowments are e1 = (4, 2) and e? = (6, 6) And their preferences are represented by utility functions: u(x, y) = x³y and u(x,y) = x³y$ (d) Set up the utility maximization problem for each consumer and solve for their Marshallian demand functions. (e) Compute the market demand for each good. () State the Walrus law for this economy and explain its economic interpretation. (g) Assume the excess demand for good x is zero, i.e., EDx = 0, and calculate the ratio of prices, i.e., p Ipy . Then, use this ratio of prices to show that the excess demand for good Yis also zero, i.e., EDy= 0. Briefly explain how this relates to the Walrus' law. (h) Given the price ratio found above, calculate the equilibrium allocations and show that feasibility, individual rationality, and Pareto efficiency holds.Consider an general equilibrium endowment economy with two goods and two individuals. The two individuals A and B have the following endowments {(w^, w), (wP, w? )} = {(4,3), (2, 1)} The utility functions for A and B are: U^(*f, a) = 0.5 log(x†)+log(x;) U® x? ,a?) = log(x)+0.5 log(xž) ** Part a (10 State the market clearing conditions. ** Part b ( Draw the Edgeworth box, including: • the x-y coordinates of the 4 corners. • the x-y coordinates of endowment point. • a sketch of some indifference curves for both individuals. ** Part c (10 Given the relative price p > 0, find the demand of good 1 for both individuals.
- Consider an exchange economy with two consumers (A and B) and two goods (x1 and x2). Consumer A has an endowment of 5 units of x1 and none of x2, whereas Consumer B has an endowment of 3 units of x1 and 15 units of x2. Consumer A's utility function is given by uA=xA1xA2, and Consumer B's utility function is given by uB=min{xB1,xB2}. Both goods are traded in competitive markets. Find the competitive equilibrium price for x2, assuming p1=1.Consider an exchange economy with two agents (i = a, b) and two goods (j = 1, 2) with the following characteristics. Agent a: Initial endowment (ea,1, Ca,2) = (1,0) and utility function Ua(Ca,1, Ca,2) = - exp(-ca,1) – exp(-ca.2) Agent b: Initial endowment (e,1, C,2) = (0, 3) and utility function U(ch1, Ch,2) = – exp(-c1) – exp(-c,2) Assume that there exists a competitive market for each good. Let p, denote the price of good j = 1,2. a) What is the competitive equilibrium of this economy? b) Illustrate the competitive equilibrium in the Edgeworth box. Draw at least the agents' indifference curves, the budget line, and the contract curve (i.e., the set of all Pareto optimal allocations). Clearly indicate which curves represent what.Suppose there are two consumers, A and B, and two goods, X and Y. Consumer A is given an initial endowment of 3 units of good X and 5 units of good Y. Consumer B is given an initial endowment of 5 units of good X and 3 units of good Y. Consumer A's utility function is given by: UA(X,Y)= X + 4Y, and consumer B's utility function is given by UB(X,Y)= MIN (X, 2Y). If the prices are Px = 1 and Py = 3, if each consumer sells her initial endowment and buys back her optimal bundle, which of the following statements is true? Group of answer choices Both Markets are in Equilibrium There is Excess Demand in the Market for Good X = 2.4 There is Excess Demand in the Market for Good Y = 0.8 There is Excess Supply in the Market for Good Y = 1.6 There is Excess Supply in the Market for Good X = 1.6 E
- Suppose there are two consumers, A and B, and two goods, X and Y. Consumer A is given an initial endowment of 2 units of good X and 6 units of good Y. Consumer B is given an initial endowment of 6 units of good X and 2 units of good Y. Consumer A's utility function is given by: UAXY) = X"Y, and consumer B's utility function is given by: Ug(X,Y) - MIN (2X, Y). If the prices are Px-1 and Py- 2, if each consumer sells her initial endowment and buys back her and in the optimal bundle, then in the Market for Good X there is Select) Market for Good Y there is [Select)In an exchange economy, two agents have utility functions u^(x, y) = x2 y and u"(x, y) = x · y, respectively, from x units of Good 1 and y units of Good 2. Assume the initial endowments are w A = (@A.1,2) and oB = (32,40), respectively. Suppose that an equilibrium is found in which the prices of the good are equal (that is, Pi = P2, i.e., the relative price is 1). If WA.1 = 60, then wA.2 %3DConsider a two-person, two good pure exchange economy. A's preferences over consumption bundles (x₁, x₂) are represented by the utility function UA (X₁, X₂) = x₁Xz B's preferences over consumption bundles are represented by the utility function UB (X₁, X₂) = x₁ + 2x₂ The initial endowment in goods x, and x₂ are given as: Individual A's as e^ = (1,0.5) and Individual B's as eB= (1, 1.5) 1. Draw an Edgeworth box indicating the endowment and preferences of this problem. 2. Find the set of Pareto Optimal Allocations in this economy and depict these in the Edgeworth box. MRSA = JUA/ƏX₁ X2 aUA/aX₂ X1 ƏUB/0x₁ MRSB = = 2 aUB/0x₂ Therefore, the set of all points where the MRS are equal are given by = 2 → X₂ = 2x1 Show Transcribed Text Advanced Microeconomics. Solve step by step so that I grasp the concept
- Problem 1: Exchange economy Consider an exchange economy with two consumers, A and B, and two goods, X and Y. Consumer A has an initial endowment TA = i >0 of good X, an initial endowment yA = 0 of good Y, and preferences over consumption bundles that can be represented by the Cobb-Douglas utility function uA(TA: YA) = ", where ra is the quantity of good X, yYA is the quantity of good Y, and a € (0, 1) is a preference parameter. Consumer B has an initial endowment ig = 0 of good X, an initial endow- ment js = j > 0 of good Y, and preferences over consumption bundles that can be represented by the Cobb-Douglas utility function up(rB, YB): B1-3 where rB is the quantity of good X, YB is the quantity of good Y, and BE (0, 1) is a preference parameter. (a) Find the set of Pareto optimal allocations in this economy. In appro- priate diagrams, illustrate the set of Pareto optimal allocations when (i) 3 = a, (ii) 3 > a, and (iii) 3 a.An exchange economy consists of two individuals and two goods. The two individuals have the following Leontief utility functions: Person 1: U1(x1, y1) = 3x1 + y1 Person 2: U2(x2, y2) = x2 + 2y2 Person 1 has an endowment of e1 = (3, 2). Person 2’s endowment is e1 = (3, 4). In an Edgeworth Box diagram, show which allocations are in the core. Describe 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). What price ratio(s) P* will support a competitive equilibrium allocation for this economy?Consider an economy with two consumers A and B, one firm and two goods, 1 and 2. The initial endowments of A and B are w = wB = (1/2, 1/2). The utility functions are: u^(rf, 12) = In z{' + In r and "(f.) = Vzf + V=? The firm produces good 2 using good 1 as input, the production function is y2 = The consumer B owns the firm (denote z the firm's profit). Good 2 is the numeraire good (i.e., p2 = 1). (a) Determine the demand for good 1 of the consumers and the firm. (b) Show that there is a unique equilibrium price p. (c) Prove or disprove that the final allocation is Pareto Optimal.