6. (b) Consider an exchange economy with two agents Á and B, and two goods i and y. Consumption of the two goods is given by: „(1-a) U^(Xa»Ya) = xfyt-a) U°(Xp,Y») = x$y{t-b) Where xA is agent A's consumption of good x etc. Let the agents' initial endowments be: Good Agent A 1 B 1 (%)- i) Find competitive equilibrium price ratio |for this economy. ii) Why is it sufficient to compute the market clearing price ratio for just one ma (say) to answer (i).
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- 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.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.Give typing answer with explanation and conclusion Consider an exchange economy consisting of two people, A and B, endowed with two goods, 1 and 2. Person A is initially endowed with ωA = (0,10) and person B is initially endowed with ωB = (11,0). They have identical preferences, which are given by U^A(x1,x2) = U^B(x1,x2) = x1^2*x2. Suppose that p2 =1. Under the competitive equilibrium, what is p1? Round answers to two decimal places.
- 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?Throughout this problem set, we will look at exchange economies with two goods and two agents. Let X = R², let u denote agent i's utility, and let wie X denote agent i's endowment. 1. Suppose u¹(x¹) = min{ri, 2} and wi = (4,8) for both agents i. (a) Argue that every Pareto optimal allocation has r≥r for both agents i. (b) Argue that every allocation z with r≥r for both agents i is Pareto optimal. (c) Draw an Edgeworth box, with a picture depicting every Pareto-optimal allocation. In this picture, also draw the endowment allocation, and draw each agent's indifference curve through the endowment. (d) Argue that, in any competitive equilibrium, the price of good 2 must be zero. (e) Find all competitive equilibria.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 %D
- Consider a pure-exchange economy of two individuals (A and B) and two goods (X and Y). Individual A is endowed with 5 units of good X and 3 units of good Y, while individual B with 3 and 4 units of goods X and Y, respectively. Assuming utility functions of individuals A and B to be UA=XA YA² and UB-XB² YB where Xi and Yi for i= {A, B} represent individual i's consumption of good X and Y respectively, what will be the set of Pareto optimal allocation in this economy?There are two consumers, i = 1, 2. There are L traded goods in the economy and the consumers are price takers. Each consumer has preferences over the commodities she consumes and over some action h that is taken by consumer 1. That is, Ui (xị, ..., x, , h) Activity h is something that has no direct monetary cost for person 1. For example, it could be playing loud music. From the point of view of consumer 2, h represents an externality of consumer 1's actions if = 0 +0 04) 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 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 concept3 Consider a pure exchange economy with 2 consumers and 2 goods. Consumer ? owns 8 units of good 1 and 1 unit of good 2, and his preference is represented by the following utility function: uA(x1, x2) = x1*x2. Consumer ? owns 2 units of good 1 and 4 units of good 2, and his preference is represented by the following utility function: uB(x1,x2)=x1+x2. Assume that the two consumers are allowed and able to trade with each other, and that good 1 is the numeraire. In this case, both consumers act as price-takers. Price takers: a market participant that is not able to dictate the prices in a market. Therefore, a price taker must accept the prevailing market price. Please find the competitive equilibrium of this pure exchange economy.4. Consider a two-consumer, two-good exchange economy. Utility functions and endowments are u'(x1, x2) = (x1x2)² and e' = (18, 4),u²cx1, x2) = In(x1) + 2 In(x2) and e? = (3, 6). (a) Characterize the set of Pareto-efficient allocations as completely as possible. (b) Characterize the core of this economy. (c) Find a Walrasian equilibrium and compute the WEA. (d) Verify that the WEA you found in part (c) is in the core.