Consider a two-good, two-consumer pure exchange economy of the following data:

U A(x A 1 , xA 2 ) = (x A 1 ) a (x A 2 ) 1?a , wA = (1, 0)

U B(x B 1 , xB 2 ) = (x B 1 ) b (x B 2 ) 1?b , wB = (0, 2)

where a, b ? (0, 1).

2.1 Find the competitive equilibrium prices and allocation(s) of this economy.

2.2 Compute the utilities that consumer A and consumer B receive at the equilibrium allocation. Compare them to the utilities each would receive if they simply consumed their initial endowment. Using this information to answer the two questions: Does the competitive equilibrium make both better off? Was the initial endowment Pareto efficient?

2.3 How do we know the equilibrium found in 2.1 is Pareto efficient? One way to know it is to solve the problem of maximizing A's utility subject to the resource constraints and the constraint that B receives the utility level found in 2.2. Formulate the optimization problem, reduce it to a problem with two choice variables and one constraint, then formulate the Lagrange function for the resulting problem, and show that the solution is the same found in 2.1. Explain why this way gives you a Pareto efficient allocation.

2.4 An alternative way to know the equilibrium found in 2.1 is Pareto efficient is to solve the problem of maximizing B's utility subject to the resource constraints and the constraint that A receives the utility level found in 2.2. Show that this gives the same allocation. Explain why the two ways must give the same allocation.

2.5 You have just verified the first welfare theorem. To see the second theorem, identify a Pareto efficient allocation different from the competitive equilibrium. Show that this allocation is indeed Pareto efficient. Then, find the endowments for which this new allocation is a competitive equilibrium in this new economy (remember that initial endowments are part of the data of the economy).

2 General equilibrium analysis In our lectures, we discussed the relationship between competitive equilibrium out come and Pareto efciency. In what follows, let's explore some issues along this relationship. Consider a twogood, twoconsumer pure exchange economy of the following data: UAlmf1$i=ir\"{$)'\" \"w" = (1:0) Ulmigmgl = {millbirilll'l'a w\" = (0,?) where 11,5 E [I], 1). 2.1 Find the competitive equilibrium prices and allocation{s) of this economy. 2.2 Compute the utilities that consumer A and consumer B receive at the equilibrium allocation. lCompare them to the utilities each would receive if they simply consumed their initial endowment. Using this information to answer the two questions: Does the competitive equilibrium make both better off? Was the initial endowment Pareto eicient? 2.3 How do we know the equilibrium found in 2.1 is Pareto efficient? One way to know it is to solve the problem of maximizing A's utility subject to the resource constraints and the constraint that B receives the utility level found in 2.2. Formulate the optimization problem, reduce it to a problem with two choice variables and one constraint, then formulate the Lagrange function for the resulting problem, and show that the solution is the same found in 2.1. Explain why this way gives you a Pareto efficient allocation. 2.4 An alternative way to know the equilibrium found in 2.1 is Pareto efficient is to solve the problem of maximizing B's utility subject to the resource constraints and the constraint that A receives the utility level found in 2.2. Show that this gives the same allocation. Explain why the two ways must give the same allocation. 2.5 You have just verified the first welfare theorem. To see the second theorem, identify a Pareto efficient allocation different from the competitive equilibrium. Show that this allocation is indeed Pareto efficient. Then, find the endowments for which this new allocation is a competitive equilibrium in this new economy (remember that initial endowments are part of the data of the economy)