Problem 3: Consider a pure exchange economy with two consumers (A 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 A 81-8 = 0 < a <1 0 < 3 <1 XBYB a. Derive the equation of the contract curve. b. Let the endowments of the economy be 1 unit of Xand 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?
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- Sarah 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 %D4) 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.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.
- 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.In 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)Remy and Emile consume only blueberries (x,) and raspberries (x,). Remy has utility function UR = x (x5)² and Emile has utility function UE = (xf)²x. Remy is endowed with 5 blueberries and 5 raspberries and Emile is endowed with 10 blueberries and 10 raspberries. (a) Derive the equation of the contract curve, i.e., find x%(xf). (b) Set x2 as the numeraire, i.e., assume p, = p and p2 = 1. Find the competitive equilibrium – the equilibrium price ratio, P1/P2, and the equilibrium allocation, ((xf, x5), (xf,x£ )).
- 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.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 an exchange economy with 2 goods (1 and 2) and 2 consumer (A and B). a bundle with x units of good 1 and y units of good 2 is written as (x,y). consumer A has an endowment (4,0) and consumer B has an endowment (12,12). the 2 goods are perfect substitutes for each consumer. consider an allocation in which A receives (1,9) and B receives (15,3) if we can redistribute endowments suitably, it is possible to obtain this allocation as the outcome of a competitive equilibrium. is this true or false? explain carefully
- 1. Consider a pure exchange economy with two goods and two consumers. Let F denote food and C denote clothing. Lacy has the utility function U(F, C) = F¹/32/3. Roy has the utility function V (F, C) = F2/3¹/3. Each consumer has an initial endowment consisting of 9 units of F and 9 units of C. Normalize the price of F to one. Let P denote the price of C. (a) Is the initial endowment a Pareto efficient allocation of F and C between the two con- sumers? Explain briefly. (b) What is each consumer's demand for F and C as a function of P? [Hint: the wealth of each consumer is 9 + 9P.] (c) What is the price of C in a competitive equilibrium? (d) What is the allocation of F and C between the two consumers in a competitive equilibrium?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). AssumingQ. 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.