Consider the following pure exchange economy with different consumer pref- erences and endowments (e.g., wA = (w, w)), and answer the following ques- tion: (a) Draw the Edgeworth box and the endowment point. (b) Find the contract curve yA = f(xA), and plot it on the graph. (c) Find the Pareto optimal allocation and price ratio for the corresponding competitive equilibrium.
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- Problem 3: Consider a pure exchange economy with two consumers (4 and B) and two goods (X and Y). Suppose consumers A and B have the utility functions: UA (XA, YA) u³ (XB, YB) = a1-a 81-8 = XBYB a. Derive the equation of the contract curve. b. Let the endowments of the economy be 1 unit of X and 2 units of Y. Show that an equal division of goods (an egalitarian outcome) is not Pareto optimal. What condition on the parameters of the utility functions is required for equal division of goods to be Pareto optimal? 0 < a < 1 0 < 3 <1 3Sarah 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 %DIn an exchange economy, there are two people (Shadi and Nino) and two goods (x1 and x2). Their initial endowments are ωS = (2, 4) and ωN = (3, 6). Their utility is given by the following functions: US(x1,x2) = x12x23 and UN(x1,x2) = x1x24. Which of the following is the equation for the contract curve? Group of answer choices a. x2N = 96x1N / (15 + 4x1N) b. x2N = 47x1N / (8 + 4x1N) c. x2N = 91x1N / 5 d. x2N = 16x1N / (3 + x1N) e. x2N = 41x1N / (9 + x1N)
- 1. Consider the following pure exchange economy consisting of two consumers with the following utility functions and initial endowments: uz (X11, X12) = X11 +x12 uz(X21, X22) = x21 + X22 W1 = (3,1) W2 = (1,3) a. Find the set of Pareto optimal allocations, PO(Ep). (12p) b. Find the set of individually rational allocations, IR(Ep). (12p) c. Find the set of core allocations, C(Ep). (10p)Is the following statement correct: "For any two-person exchange economy, if each consumer has a complete, transitive and convex preference relation, then every competitive equilibrium allocation is efficient"? Provide a proof of your argument. [For this question, maximum word limit is 80.]Consider a two-people, two-goods exchange economy, where Person A and Person B have the following utility functions. In which case, the Pareto optimal allocation can be on the boundary of the Edgeworth box? Note: Do not consider the origins as the boundary points. (Boundary points are the edges of the Edgeworth box except for the origins.) 0 UA = min (XA1, XA2) and Ug = Xg1.Xg2 (i) UA = log(xA1) + log(Xa2) and ug = 3Xgt + X82 (ii) uA = (XA1) 0XA) and ug = 3log(Xgy) + X82 %3! (iv) uA = (XA1)2 + XA2 and ug = log(xa1) + Xe2
- 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.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). AssumingSuppose that 2 agents, A and B, have preferences over goods 1 and 2: u+(x4) 2x + x5 and u (x³) = xFx. Total endowments are ej = B B Total endowments are e¡ = e2 = 4, (a) Find the contract curve. (b) If initial endowments are e4 = A (4,0) and eB (0, 4), what is the core?
- (4) Resource Allocation. Consider a combinatorial auction with three items {a,b,c} and three players (I, II, III. The valuations each player has for each subset of a b с ab bс ас abc I 1 2 1 6 4 11 II 3 5 3 3 5 III| 4 5 0 7 5 7 the items is shown below. What is the allocation produced under the VCG (Vickrey-Clarke-Groves) mechanism and what are the VCG payments of the players?Consider trade between two consumers (1 and 2) and two goods, X and Y. Suppose the total quantities of each good are 100 units. Each consumer has Cobb-Douglas preferences given by: U(X,Y) = XY Denoting by X1 the first consumer's consumption of X and by Y1 the first consumer's consumption of Y, the contract curve consists of all allocations where A. X1 = 0.5Y1 B. X1 = 2Y1 C. X1 = Y1 square D. X1 = Y1Problem 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.