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(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

x

and

Y

is denoted

(x^(**),Y^(**))

. A's preferences are represented by the utality function

U^(**)

where

U^(A)(xA,YA)=xAYA

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

U^(B)(xB,YB,xA)=xBYB-CxA

where

xB

and

YB

denote

B

's consumption of goods

x

and

Y

, respectively, and

C

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

2n

persons with

n

of them being exact clones of A and the remaining

n

being exact clones of

B

, that both

A

and

B

have nonnegative endowments of the two goods, denoted

(x\lambda ,Y\lambda )

and

(x\beta ,Yb)

, respectively, with

xA^(?) xB^(?)=x^(**) and YA^(?) YB^(?)=Y^(**)

and that each B type of consumer has preferences given by

U^(B)(xB,YB,xA)=xBYB-CxA

where

\angstrom A

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.



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