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4. Suppose A and B are in a pure exchange economy with no production. The utility function of Abe is given as $u_A = x_A^{1/2}y_A^{1/2}$ and the utility function of Betty is given as $u_B = x_B^{1/3}y_B^{2/3}$. Suppose the following initial endowment situation: Abe's endowment = (1, 1) and Betty's endowment = (1, 1). (a) Is the initial endowment Pareto optimal (efficient)? Why or why not? (b) Find a function representing the set of Pareto efficient allocations, and draw it on an Edgeworth box. (c) Show that the competitive equilibrium allocation is Pareto efficient.

          4. Suppose A and B are in a pure exchange economy with no production. The utility function of Abe is given as $u_A = x_A^{1/2}y_A^{1/2}$ and the utility function of Betty is given as $u_B = x_B^{1/3}y_B^{2/3}$. Suppose the following initial endowment situation: Abe's endowment = (1, 1) and Betty's endowment = (1, 1).
(a) Is the initial endowment Pareto optimal (efficient)? Why or why not?
(b) Find a function representing the set of Pareto efficient allocations, and draw it on an Edgeworth box.
(c) Show that the competitive equilibrium allocation is Pareto efficient.

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4. Suppose A and B are in a pure exchange economy with no production. The utility function of Abe is given as uA = xA^1/2yA^1/2 and the utility function of Betty is given as uB = xB^1/3yB^2/3. Suppose the following initial endowment situation: Abe's endowment = (1, 1) and Betty's endowment = (1, 1).
(a) Is the initial endowment Pareto optimal (efficient)? Why or why not?
(b) Find a function representing the set of Pareto efficient allocations, and draw it on an Edgeworth box.
(c) Show that the competitive equilibrium allocation is Pareto efficient.

Added by Jasmine Q.

Principles of Economics

Gregory Mankiw 8th Edition

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Solved by Expert Banhishikha Sinha

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In this case, the initial endowment is (1, 1) for both Abe and Betty. Let's consider an alternative allocation where Abe has (2, 1) and Betty has (1/2, 1). For Abe, the utility in the initial endowment is uA = (1)^(1/2)(1)^(1/2) = 1. In the alternative Show more…

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Suppose A and B are in a pure exchange economy with no production. The utility function of Abe is given as uA = x^(1/2)y^(1/2) and the utility function of Betty is given as uB = x^(1/3)y^(2/3). Suppose the following initial endowment situation: Abe's endowment = (1, 1) and Betty's endowment = (1, 1). (a) Is the initial endowment Pareto optimal (efficient)? Why or why not? (b) Find a function representing the set of Pareto efficient allocations, and draw it on an Edgeworth box. (c) Show that the competitive equilibrium allocation is Pareto efficient.

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