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Utility
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Marginal Rate of Substitution
The MRS is the slope of the indifference curve at a point
MRS=derivative of indifference curve
2x
1x
),( 21 xx
1x
2x
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Interpretation of MRS
The MRS measures the rate at which the consumer is willing (i.e., indifferent) to substitute one good for the other.
If good 2 is measured in dollars, the MRS measures the consumer’s willingness to pay for an extra unit of good 1.
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Assumptions on Preferences and the MRS
Monotonicity: MRS negative
(Strict) Convexity: MRS decreases as
increases
2x
1x1x
2x1x
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Utility Function
Idea: assign a number to each consumption bundle, with higher numbers assigned to more-preferred bundles.
A utility function represents a preference relation :
),(),(),(),( 21212121 yyuxxuyyxx
),( 21 xxu
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Utility Function: Does It Always Exist?
Q: Given a preference relation can we find a utility function that represents it?
A: If preferences are complete and transitive (plus a technical assumption called “continuity” is verified) we can. [Sufficient condition]
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Is Transitivity Necessary?
Q: Is it necessary that preferences are transitive for the existence of a utility function that represents them?
A: Yes. Otherwise:
)1,1()1,4.0()5.0,1()1,1(
)1,1()1,4.0()5.0,1()1,1( uuuu
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Utility is Just Ordinal
E.g.: (1,1) (1,0.5) (0.4,1) (0.8,0.8)
Can be represented in different ways:
Bundle
(1,1) 3 6 -1
(1,0.5) 2 4 -2
(0.4,1) 1 2 -3
(0.8,0.8) 1 2 -3
),( 21 xxu ),(2 21 xxu 4),( 21 xxu
~
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In General
If :
1) represents
2) is a positive monotonic transformation
Then, also represents
),( 21 xxu )(uf
)),(( 21 xxuf
)]()([ 2121 ufufuu
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Utility Functions and Indifference Curves
2x
1x
Utility function:
Indifference curves:
or:
2121 ),( xxxxu
uxx 21
12x
ux 2
3
5.12x
1x
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Indifference Curves and Monotonic Transformations
2x
1x
Utility function:
Indifference curves:
or:
22
2121 ),( xxxxv
12
x
vx 4
925.1
2x
1x
vxx 22
21
2y
1y
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Perfect Substitutes
2x
1x
Utility function:
Indifference curves:
2121 ),( bxaxxxu
12 xb
a
b
ux
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Perfect Complements
2x
1x
Utility function:
),min(),( 2121 bxaxxxu
12 xb
ax
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Cobb-Douglas Preferences
2x
1x
Utility function:
Indifference curves:
dc xxxxu 2121 ),(
d
c
d
x
ux
1
1
2
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Cobb-Douglas
Most commonly used utility function in economics
Possible to assume without loss of generality that
Why?: apply to get
1 dc
dcuuf 1
)(
dc
d
dc
c
xxxxv 2121 ),(
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Marginal Utility
Consider a consumer that is consuming
Q: By how much does his utility change as we increase by a very small amount his consumption of good 1?
A: Marginal utility of good 1:
),( 21 xx
1
2,1 )(
x
xxu
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Marginal Utility and Units
The marginal utilities of good 1 and good 2 depend on the specific utility function we are using.
Consider:
Then:
))((),( 2,121 xxufxxv
1
2,12,1
1
2,1 )())(()(
x
xxu
u
xxuf
x
xxv
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Marginal Utility and MRS
MRS only depends on preferences and not on their specific representation (utility function).
Marginal utilities can be used to compute the MRS between two goods.
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Computing the MRS
Consider an indifference curve:
Q: What is the slope of this indifference curve (MRS)?
A:
uxxu ),( 21
2
2,1
1
2,1
1
22,1
)(
)(
)(
xxxu
xxxu
x
xxxMRS
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The MRS and Monotonic Transformation
Consider a monotonic transformation
Q: What is the MRS in this case?
A:
)(2),( 2,121 xxuxxv
2
2,1
1
2,1
2
2,1
1
2,1
2,1)(
)(
)(2
)(2
)(
xxxu
xxxu
xxxu
xxxu
xxMRS