Homework 4, Spring 08
(Total 20 points), Due on Mar. 11, Thu.
Q1
(7pt.): Consider an exchange economy with two goods and
n
individuals,
in which consumer
i
’s preference is represented by
u
i
(
x
i
1
, x
i
2
) =
a
i
√
x
i
1
+
b
i
√
x
i
2
and her initial holding is given by (
ω
i
1
, ω
i
2
), where
i
= 1
,
···
, n
.
Establish the equation which determines the competitive equilibrium relative
price
p
*
1
p
*
2
. — You don’t have to solve it.
Answer
: For each
i
, her MRS is
a
i
2
√
x
i
1
b
i
2
√
x
i
2
=
a
i
√
x
i
2
b
i
√
x
i
1
Hence the tangency condition
MRS
=
p
1
p
2
yields
x
i
2
=
b
2
i
p
2
1
a
2
i
p
2
2
·
x
i
1
Plug this into the budget equation
p
1
x
i
1
+
p
2
x
i
2
=
p
1
ω
i
1
+
p
2
ω
i
2
, we have
x
i
1
=
p
1
p
2
ω
i
1
+
ω
i
2
p
1
p
2
+
a
2
i
b
2
i
·
p
2
1
p
2
2
,
x
i
2
=
b
2
i
p
2
1
a
2
i
p
2
2
·
p
1
p
2
ω
i
1
+
ω
i
2
p
1
p
2
+
a
2
i
b
2
i
·
p
2
1
p
2
2
The equation for the equilibrium price boils down to
∑
i
x
i
1
=
∑
i
ω
i
1
, hence
it is
±
i
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 Spring '10
 d
 Economics, Supply And Demand, 2pt, 7pt, 3pt, Bevo Chemical, Co.

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