(Solved): Consider a simple exchange economy with two individuals, A and B , and two gosds, X and Y . The econ ...

Consider a simple exchange economy with two individuals, A and B , and two gosds, X and Y . The economy's endowment of each good is positive and the total endosmert of goods

and

is denoted

. A's preferences are represented by the utality function

where

and XA and YA denote A's consumption of goods X and Y, respectively, B's preferences are represented by the utility function

where

and

denote

's consumption of goods

and

, respectively, and

is a positive parameter. a) Determine the Pareto optimal allocations for this economy. Recall that one way of describing a Pareto optimal allocation when there are two persons is that the allocation maximites the utility of person 1 given a fixed utility level for person 2. Use that in deriving the Pareto optimal allocasions. b) Suppose that the economy has a total of

persons with

of them being exact clones of A and the remaining

being exact clones of

, that both

and

have nonnegative endowments of the two goods, denoted

and

, respectively, with

and that each B type of consumer has preferences given by

where

is the mean X consumption of the A type coesumers. Determine the competive. equilibrium allocations and the corresponding equilibrium prices. c) Determine whether the competitive equilbrium of the economy in part (a) is Partte sotimal.