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Question 1 Suppose a consumer's utility function is given by U(x,,x) = x, x2

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Question 1 Suppose a consumer's utility function is given by U(x,,x) = x, x2. The price of good 2 is given by P, = 1 and income M = 12. - Find the optimal bundle that maximizes the consumer's utility in three different price situations - Price of good 1: P, = 4 - Price of good 1: P, 3 Price of good 1:p = 1 Plot the optimal bundles in the three cases found above to obtain the price offer curve (set good 1 on the x-axis and good 2 on the y-axis) Question 2 A consumer's utility function is given by U(x,,*) = min{xy, x,} with the price of good 2 as p, = 1 and income M = 12. Find the optimal bundle that maximizes the consumer's utility in the following three cases: - Price of good 1:24 = 4 - Price of good 1:24 = 3 Price of good 1:2, = 1 Plot the optimal bundles in the three cases found above to obtain the price offer curve (set good 1 on the x-axis and good 2 on the y-axis) Question 3 Suppose there are two individuals in a market: Agent A and Agent B. Their utility function for goods X and Y are given by U*(x, y) = x^y^ and U?(x, y) = x®y, respectively. Agent A and Agent B each have the same endowments w^ = (2, 2) and w' = (2, 2). Find the equations for the Pareto Optimal (PO) allocations in this economy, and show them in an appropriately-labeled Edgeworth Box B Question 4 - 4A: With respect to the Edgeworth Box diagrams presented in lecture, what is the significance of the contract curve? Can trade/exchange take place at any location on the Pareto Efficiency Locus? 4B: Under Exchange via Negotiation we considered a replica economy of Agent A, Agent B, and the respective twin agents for A and B. One twin proposed a reallocation scheme where each receives half of each good that they have at that given point. Would this group of Agent A, Agent A's twin, and Agent B be better off, where Agent B will not be worse off? Explain with appropriate work showing the resulting allocation and preferences for the proposal. - In this replica economy now with 4 individuals, what happened to the number of Pareto efficient allocations compared to the economy with only 2 individuals? What happens to the number of possible allocations as the economy replicates iteratively?