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Consider a two-person, two-good economy (say apples and bananas) without production

Math

Consider a two-person, two-good economy (say apples and bananas) without production. Both consumers have identical preferences, given by u(a,b)= a^0.2 * b^1-0.2 , where a denotes the consumption of apples, and b the consumption of bananas. Person 1 owns 40 bananas and 20 apples, whereas person 2 owns 20 bananas and 40 apples. Calculate the equilibrium value of P2/P1 for this economy.
4. For this question, refer back to data from question 3. Suppose the government of this economy adopts a Rawlsian approach to welfare economics. That is, the social welfare function they adopt is as follows: S(u1, u2)= min {u1,u2} Calculate the Socially Optimal allocation for this economy. Can such an allocation be achieved via re-distribution of the endowments?
5. For this question, refer back to the data on question 3. Suppose that some brilliant scientists find a way to transform 1 banana into two apples or, by reversing the process, 2 apples into 1 banana. Formally, the production set is as follows: Y= {(a,b): a ≤ -2b)} Where is a and denote (respectively) the units of apples and bananas employed as inputs/outputs. How many bananas will be produced in equilibrium? Note: if bananas are used as an input to produce apples, then your answer should be a negative number. Hint: you don't need to produce many calculations. Simply ask yourself what will the relative prices in this economy be? Then, how does the market clearing condition look like? What is the solution for the number of bananas?

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