Problem:
This question is about Walrasian equilibrium in an exchange economy with 2 goods and 2 consumers. Taxes are introduced in the question to solve for the equilibrium and allocation under Pareto theorem.
Question:
Consider an exchange economy with 2 goods and 2 consumers . Consumer 1's initial endowment is and consumer 2's endowment is . In each case , the first entry in the endowment vector denotes the initial endowment of good 1, and the second entry the initial endowment of good 2 . Both consumers have the same consumption set: , and the same utility function: Suppose that consumer 1's expenditure on good 1 is taxed at a rate of 50%, and that the revenue from this tax is paid as a lump-sum transfer to consumer 2.
Find the Walrasian equilibrium of this economy. Then find another allocation which Pareto-dominates the equilibrium allocation.