Suppose that Anne and Bart can trade with each other. Anne's utility function is UA(xA, YA)...

 

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Suppose that Anne and Bart can trade with each other. Anne's utility function is UA(xA, YA) = min (2x4, YA) and Bart's utility function is UB(XB, YB) = (2xB,YB), where x4 and Ya are Anne's consumption of x and y, while g and yg are Bart's consumption of x and y. Anne is endowed with 10 of x and 5 of y, and Bart is endowed with 0 of x and 10 of y. a) Draw an Edgeworth box of this economy of Anne and Bart, labeling the initial endowment point e, with Anne and Bart's indifference curves going through e. (3 pts.) b) What are the values of MRS at the endowment point e, and does it tell you anything about the pareto efficiency of point e? If so, what does it say? If not, why not? (3 pts.) c) Clearly define, using set notations, the allocations that are pareto efficient. (4 pts.) d) What are the utility levels for Anne and Bart at point e? (2 pts.) e) If the price of x is 1 and y is 2, what is Anne's optimized consumption? (4 pts.) f) If the price of x is 1 and y is 2, what is Bart's optimized consumption? (4 pts.) g) Prove that your answers from e) and f) are incompatible. In other words, show how the markets of x and y are not going to clear if the price of x is 1, and y is 2. (4 pts.) h) At what price ratio (or prices) would they voluntarily trade and optimize their consumptions, and what is the optimized consumption for each person? (6 pts.) Suppose that Anne and Bart can trade with each other. Anne's utility function is UA(xA, YA) = min (2x4, YA) and Bart's utility function is UB(XB, YB) = (2xB,YB), where x4 and Ya are Anne's consumption of x and y, while g and yg are Bart's consumption of x and y. Anne is endowed with 10 of x and 5 of y, and Bart is endowed with 0 of x and 10 of y. a) Draw an Edgeworth box of this economy of Anne and Bart, labeling the initial endowment point e, with Anne and Bart's indifference curves going through e. (3 pts.) b) What are the values of MRS at the endowment point e, and does it tell you anything about the pareto efficiency of point e? If so, what does it say? If not, why not? (3 pts.) c) Clearly define, using set notations, the allocations that are pareto efficient. (4 pts.) d) What are the utility levels for Anne and Bart at point e? (2 pts.) e) If the price of x is 1 and y is 2, what is Anne's optimized consumption? (4 pts.) f) If the price of x is 1 and y is 2, what is Bart's optimized consumption? (4 pts.) g) Prove that your answers from e) and f) are incompatible. In other words, show how the markets of x and y are not going to clear if the price of x is 1, and y is 2. (4 pts.) h) At what price ratio (or prices) would they voluntarily trade and optimize their consumptions, and what is the optimized consumption for each person? (6 pts.)

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