1
ECON191 (Spring 2009)
2324.2.2009 (Tutorial 3)
Chapter 2 Theory of Consumer
(Chapter 3 Consumer Behavior)
Marginal rate of substitution
±
)
,
(
y
x
U
U
=
Ö
y
y
U
x
x
U
U
Δ
∂
∂
+
Δ
∂
∂
=
Δ
Ö
(
x
MU
U
x
U
MU
x
x
Δ
=
Δ
⇒
Δ
Δ
=
and
y
MU
U
y
U
MU
y
y
Δ
=
Δ
⇒
Δ
Δ
=
)
Ö
0
=
Δ
+
Δ
=
Δ
y
MU
x
MU
U
y
x
Ö
y
MU
x
MU
y
x
Δ
−
=
Δ
Ö
y
x
MU
MU
x
y
=
Δ
Δ
−
and
x
y
MRS
xy
Δ
Δ
−
=
Ö
y
x
xy
MU
MU
MRS
=
The Utility maximization
±
Bundle A, B, C, D and E are on the BL, income
is exhausted. Bundle F does not use up the whole
income.
±
If we are at bundle A, with the given income
Ö
Utility increases if we move to C and utility is
maximized if we move to E. (Increase the
consumption of X and reduce Y)
±
At point E, Slope of IC = slope of BL.
Ö
Maximization condition:
Y
X
Y
X
P
P
MU
MU
MRS
=
=
Ö
Y
Y
X
X
P
MU
P
MU
=
(meaning?)
IC
2
IC
1
E
Y
X
B
A
IC
3
D
C
F
Budget line: slope = P
x
/ P
y
X
*
Y
*
±
Marginal rate of substitution (MRS)
: Maximum amount of
a good that a consumer is willing to give up in order to
obtain one additional unit of another good.
±
Marginal Utility
: additional satisfaction obtained from
consuming one additional unit of a good.
Ö
x
y
MRS
xy
Δ
Δ
=
y
x
MU
MU
=
±
Law of diminishing marginal rate of substitution
Ö
From A to B, the slope becomes flatter.
X
IC
1
A
Y

Δ
Y
Δ
X

Δ
Y
Δ
X
B
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Example
:
Assume that there are two goods in the world: apples and raspberries. Say that Geoffrey has a
utility function for these goods of the following type, where
r
denotes the quantity of
raspberries and
a
the quantity of apples.
ra
U
=
a)
Draw the indifference curve that is defined by the utility function and has a utility level of
2500.
b)
What is the marginal rate of substitution between raspberries and the apples when
Geoffrey consumes 50 raspberries and 50 apples? What is the marginal rate of
substitution between these two goods when Geoffrey consumes 100 raspberries and 50
apples?
r
a
MU
MUr
MRS
a
ra
=
=
.
MRS
ra
when Geoffrey has 50r and 50a =
1
,
MRS
ra
when Geoffrey
has 100r and 50a =
2
1
c)
If the price of raspberries is $1 per unit and the price of apples is $1 per unit and Geoffrey
has $100 to spend, what bundle of raspberries and apples would he buy? Is the marginal
rate of substitution be equal to the ratio of the prices of these goods in the optimal bundle?
If not, why not?
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 Spring '08
 Chen
 Microeconomics, Marginal rate, X1, optimal bundle, Geoffrey MU

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