Consider a small exchange economy with two consumers, A and B, and two commodities, 2₁ and ₂. Consumer A's initial endowment is 3 units of ₁ and 2 unit of 2. Consumer B's initial endowment has 5 units of 2₁ and 4 units of ₂. Consumer A's utility function is given by U(r₁, ₁) = x₁x₁. Consumer B's utility function is given by U(x,x) = 2x² + x2.
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- 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.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?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 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 conceptConsider an exchange economy with two consumers: Charlotte and Dylan, and two goods: quinoa (Q) and raspberries (R). Charlotte has an initial endowment of 83.9 units of quinoa and 99.3 raspberries. Dylan has 112.5 units of quinoa and 77.8 raspberries. Charlotte's utility function is given by Uc=Q¹/² R¹/², where Qc and Re are her consumption of Q and R, respectively. Dylan's utility function is given by Up-Q¹/³ R2/3, where Qp and Rp are his consumption of Q and R, respectively. Suppose that the market price of quinoa is po=2 and the market price of raspberries is pr=1. At these prices, how many units of Q would Dylan want to consume?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.
- 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.● Say that v(p, y) = y ¹ - 2p₁² - 5p2. Compute the demand function 2. 2P1 -2 4. Consider a 2x2 pure exchange economy with agents 1 and 2, goods x and y, and initial endowments e¹ = (8,6) and e² = (12, 4). • Draw the Edgeworth box, identify the initial endowment and define analytically and graphically (as precisely as possible) agents' budget sets if the price vector is p = (2,4). Let agents' utility function be given by U(x, y) = x + 2y. Is the initial endowment a Pareto Efficient allocation? Is the initial endowment a competitive equilibrium? Explain your answers. . Provide the general definition of competitive equilibrium for a 2x2 pure exchange economy. ●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 the pure exchange economy with 2 goods, good 1 and good 2, and two consumers, consumer A and consumer B. Consumer A is initially endowed with 10 units of good 1 and 10 units of good 2. Consumer B is initially endowed with 2 units of good 1 and 2 units of good 2. The consumers have the following utility functions: uA(X1A,X2A)=X1AX2A²; UB(X1B,X28)=X18+X2B- Among the prices below, which ones are Walrasian equilibrium prices? O a. P1=3, p2 =2 O b. p1=4, p2 =5 O c. None of the other answers. O d. p1=5, p2 =4 O e. p1=2, P2 =34. 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.Consider a small closed economy with two consumption goods: good 1 (meat) and good 2 (berries). There are two types of agents, h and g, and they have the same preferences over consumption, represented by the utility function: u(x1, x2) = In ¤1 + In x2. However, there are twice as many type-h agents as type-g agents. The only factors of production are their labour. When a type-h agent chooses to spend a fraction a of his day producing meat and the rest producing berries then his output is (yf, y½) = (2a, 2(1 – a)). A type-g agent is more productive. When she chooses to spend a fraction B of her day producing meat and the rest producing berries then her output is (y7, vž) = (38, 12(1 – B)). The hunter-gatherers now have the possibility of opening up their economy to free trade. In world markets, 1 unit of meat can be exchanged for 2 units of berries, and the country would be a price-taker. Which of the following statements is correct? O a. Only agents of type g benefit from free trade. O…