Economics 3130
Supplementary Problems
[ The quantities of goods
x; y
and the relevant prices
p
x
; p
y
are always nonnegative real numbers]
1. Suppose that consumers 1 and 2 have the same income
m
= 1
;
and face the same prices for two goods
p
x
=
p
y
= 1
:
The utility functions of the two agents are given by:
[here
x
and
y
denote the quantities of the two goods]
u
1
(
x; y
) =
x
+
y
u
2
(
x; y
) = 13(
x
+
y
)
a) Assuming that the two consumers are maximizing their utility functions subject to the usual budget
constraint, can we predict that they will demand the
same
commodity bundle?
b) Suppose that the utility functions and income remain the same, but
0
< p
x
< p
y
:
Can we predict that
the utility maximizing commodity bundle will be the same?
In this case, derive the Engel curve showing the demand for
x
as a function of
m:
2. Index Numbers.
Suppose we observe a consumer in two di/erent periods: in the "initial" or "base"
period
b
and later in some period
t:
Let
p
b
= (
p
b
1
; p
b
2
)
and
p
t
= (
p
t
1
; p
t
2
)
be the prices of goods
1
and
2
, respectively. The consumer chooses the bundles
(
x
b
1
; x
b
2
)
and
(
x
t
1
; x
t
2
)
respectively. Assume that all
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 '06
 MASSON
 Economics, Microeconomics, Supply And Demand, pareto, 0g, budget set

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