I found in exercise 1 that the optimal bundle of good 1 and 2 for consumer A is and and for consumer B it is and . In exercise 3 I found that consumer B can block allocation y and consumer A cannot block. However, I have some trouble with the rest, and I hope I can get some help.
Let & be the following exchange economy with 2 consumers and 2 goods. The preference relations of the consumers A and B are described by the utility functions 44 and uB given by UA (2) = = 471 zz and uB(x) = 2x} - zz for all r = (21, 22) with 11, 12 > 0. Moreover, the initial endowments are wA = (3, 2) and WB = (1,4). Exercise 1. (10 %) Find the optimal bundle of every consumer i = A, B as a function of the prices p= (P1, P2) and the budget M. Exercise 2. (10 %) Find the set of all Pareto efficient feasible allocations of E and sketch them in the Edgeworth box. Exercise 3. (10 %) Let y = (y, yB) be given by yA = (2, 4) and yB = (2, 4). Show that y is a Pareto efficient feasible allocation that does not belong to the core of E and sketch the core in the Edgeworth box. Exercise 4. (10 %) Compute all competitive equilibria of E. Exercise 5. (20 %) Let z = (z, zB) be the Pareto optimal feasible allocation that is determined by z4 = 1.2. Show that (a) z = ((1.2, 4.32), (2.8, 1.68)) and (b) z is in the core of E. Show explicitly that the 3-fold replication of z, (2,2,2), is an allocation of the 3-fold replica economy of E that is not in its core. Find an allocation r of such that, for each neN, the n-fold replicated x is a core allocation of the n-fold replica economy. Exercise 6. (10 %) Consider the allocation y of Exercise 3. Find a vector of prices p and a vector T = (T4, TB) of balanced transfers so that (p, y) is a competitive equilibrium given T. Let El be the economy that arises from E by exclusively modifying the utility function of A as follows. Assume that A's preferences depend on the consumption of B as indicated by A's new utility function A defined, for each allocations r = (24, 2B), by A(x) = u4 + ^(24) 2048x Exercise 7. (10 %) Find the set of all feasible allocations that are Pareto efficient for both E and E. Exercise 8. (10 %) Find the competitive equilibrium allocation of E'. Is the statement of the 1st fundamental theorem of welfare economics valid for E'. Consider the allocation = (mA, mB) given by A = (2, 4) and B = (2,2). Exercise 9. (10 %) Show that t is Pareto efficient for E'. Is it possible to verify the statement of the 2nd fundamental theorem of welfare economics for the allocation , i.e., are there a price vector p and a balanced transfer vector T = (T4,TB) such that (p, ) becomes a competitive equilibrium of E' given T? Let & be the following exchange economy with 2 consumers and 2 goods. The preference relations of the consumers A and B are described by the utility functions 44 and uB given by UA (2) = = 471 zz and uB(x) = 2x} - zz for all r = (21, 22) with 11, 12 > 0. Moreover, the initial endowments are wA = (3, 2) and WB = (1,4). Exercise 1. (10 %) Find the optimal bundle of every consumer i = A, B as a function of the prices p= (P1, P2) and the budget M. Exercise 2. (10 %) Find the set of all Pareto efficient feasible allocations of E and sketch them in the Edgeworth box. Exercise 3. (10 %) Let y = (y, yB) be given by yA = (2, 4) and yB = (2, 4). Show that y is a Pareto efficient feasible allocation that does not belong to the core of E and sketch the core in the Edgeworth box. Exercise 4. (10 %) Compute all competitive equilibria of E. Exercise 5. (20 %) Let z = (z, zB) be the Pareto optimal feasible allocation that is determined by z4 = 1.2. Show that (a) z = ((1.2, 4.32), (2.8, 1.68)) and (b) z is in the core of E. Show explicitly that the 3-fold replication of z, (2,2,2), is an allocation of the 3-fold replica economy of E that is not in its core. Find an allocation r of such that, for each neN, the n-fold replicated x is a core allocation of the n-fold replica economy. Exercise 6. (10 %) Consider the allocation y of Exercise 3. Find a vector of prices p and a vector T = (T4, TB) of balanced transfers so that (p, y) is a competitive equilibrium given T. Let El be the economy that arises from E by exclusively modifying the utility function of A as follows. Assume that A's preferences depend on the consumption of B as indicated by A's new utility function A defined, for each allocations r = (24, 2B), by A(x) = u4 + ^(24) 2048x Exercise 7. (10 %) Find the set of all feasible allocations that are Pareto efficient for both E and E. Exercise 8. (10 %) Find the competitive equilibrium allocation of E'. Is the statement of the 1st fundamental theorem of welfare economics valid for E'. Consider the allocation = (mA, mB) given by A = (2, 4) and B = (2,2). Exercise 9. (10 %) Show that t is Pareto efficient for E'. Is it possible to verify the statement of the 2nd fundamental theorem of welfare economics for the allocation , i.e., are there a price vector p and a balanced transfer vector T = (T4,TB) such that (p, ) becomes a competitive equilibrium of E' given T
