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.