Chapter 4 Utility - ecourse2.ccu.edu.tw
Transcript of Chapter 4 Utility - ecourse2.ccu.edu.tw
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Chapter 4
Utility
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Preferences - A Reminder
• x y: x is preferred strictly to y.
• x ~ y: x and y are equally preferred.
• x y: x is preferred at least as much as is y.
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Preferences - A Reminder
• Completeness: For any two bundles x and y it is always possible to state either that x y or that y x
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Preferences - A Reminder
• Reflexivity: Any bundle x is always at least as preferred as itself; i.e. x x.
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Preferences - A Reminder
• Transitivity: If x is at least as preferred as y, and y is at least as preferred as z, then x is at least as preferred as z; i.e. x y and y z x z
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~
p
~
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Utility Functions
• A preference relation that is complete, reflexive, transitive and continuous can be represented by a continuous utility function.
• Continuity means that small changes to a consumption bundle cause only small changes to the preference level.
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Utility Functions
• A utility function U(x) represents a preference relation if and only if: x’ x” U(x’) > U(x”) x’ x” U(x’) < U(x”) x’ ~ x” U(x’) = U(x”).
p
p
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Utility Functions
• Utility is an ordinal (i.e. ordering) concept.
• E.g. if U(x) = 6 and U(y) = 2 then bundle x is strictly preferred to bundle y. But x is not preferred three times as much as is y.
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Utility Functions & Indiff. Curves • Consider the bundles (4,1), (2,3) and (2,2).
• Suppose (2,3) (4,1) ~ (2,2).
• Assign to these bundles any numbers that preserve the preference ordering; e.g. U(2,3) = 6 > U(4,1) = U(2,2) = 4.
• Call these numbers utility levels. p
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Utility Functions & Indiff. Curves
• An indifference curve contains equally preferred bundles.
• Equal preference same utility level.
• Therefore, all bundles in an indifference curve have the same utility level.
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Utility Functions & Indiff. Curves
• So the bundles (4,1) and (2,2) are in the indiff. curve with utility level U
• But the bundle (2,3) is in the indiff. curve with utility level U 6.
• On an indifference curve diagram, this preference information looks as follows:
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Utility Functions & Indiff. Curves
U 6
U 4
(2,3) (2,2) ~ (4,1)
x1
x2 p
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Utility Functions & Indiff. Curves
• Another way to visualize this same information is to plot the utility level on a vertical axis.
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U(2,3) = 6
U(2,2) = 4
U(4,1) = 4
Utility Functions & Indiff. Curves 3D plot of consumption & utility levels for 3 bundles
x1
x2
Utility
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Utility Functions & Indiff. Curves
• This 3D visualization of preferences can be made more informative by adding into it the two indifference curves.
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Utility Functions & Indiff. Curves
U
U
Higher indifference
curves contain
more preferred
bundles.
Utility
x2
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Utility Functions & Indiff. Curves
• Comparing more bundles will create a larger collection of all indifference curves and a better description of the consumer’s preferences.
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Utility Functions & Indiff. Curves
U 6
U 4
U 2
x1
x2
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Utility Functions & Indiff. Curves
• As before, this can be visualized in 3D by plotting each indifference curve at the height of its utility index.
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Utility Functions & Indiff. Curves
U 6
U 5
U 4
U 3
U 2
U 1
x1
x2
Utility
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Utility Functions & Indiff. Curves
• Comparing all possible consumption bundles gives the complete collection of the consumer’s indifference curves, each with its assigned utility level.
• This complete collection of indifference curves completely represents the consumer’s preferences.
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Utility Functions & Indiff. Curves
x1
x2
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Utility Functions & Indiff. Curves
x1
x2
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Utility Functions & Indiff. Curves
x1
x2
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Utility Functions & Indiff. Curves
x1
x2
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Utility Functions & Indiff. Curves
x1
x2
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Utility Functions & Indiff. Curves
x1
x2
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Utility Functions & Indiff. Curves
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Utility Functions & Indiff. Curves
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Utility Functions & Indiff. Curves
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Utility Functions & Indiff. Curves
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Utility Functions & Indiff. Curves
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Utility Functions & Indiff. Curves
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Utility Functions & Indiff. Curves
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Utility Functions & Indiff. Curves
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Utility Functions & Indiff. Curves
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Utility Functions & Indiff. Curves
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Utility Functions & Indiff. Curves
• The collection of all indifference curves for a given preference relation is an indifference map.
• An indifference map is equivalent to a utility function; each is the other.
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Utility Functions
• There is no unique utility function representation of a preference relation.
• Suppose U(x1,x2) = x1x2 represents a preference relation.
• Again consider the bundles (4,1), (2,3) and (2,2).
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Utility Functions
U(x1,x2) = x1x2, so U(2,3) = 6 > U(4,1) = U(2,2) = 4; that is, (2,3) (4,1) ~ (2,2). p
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Utility Functions
• U(x1,x2) = x1x2 (2,3) (4,1) ~ (2,2).
• Define V = U2.
p
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Utility Functions
• U(x1,x2) = x1x2 (2,3) (4,1) ~ (2,2).
• Define V = U2.
• Then V(x1,x2) = x12x2
2 and V(2,3) = 36 > V(4,1) = V(2,2) = 16 so again (2,3) (4,1) ~ (2,2).
• V preserves the same order as U and so represents the same preferences.
p
p
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Utility Functions
U(x1,x2) = x1x2 (2,3) (4,1) ~ (2,2).
Define W = 2U + 10.
p
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Utility Functions
• U(x1,x2) = x1x2 (2,3) (4,1) ~ (2,2).
• Define W = 2U + 10.
• Then W(x1,x2) = 2x1x2+10 so W(2,3) = 22 > W(4,1) = W(2,2) = 18. Again, (2,3) (4,1) ~ (2,2).
• W preserves the same order as U and V and so represents the same preferences.
p
p
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Utility Functions
• If
– U is a utility function that represents a preference relation and
– f is a strictly increasing function,
• then V = f(U) is also a utility function representing .
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p
~
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Goods, Bads and Neutrals • A good is a commodity unit which increases
utility (gives a more preferred bundle).
• A bad is a commodity unit which decreases utility (gives a less preferred bundle).
• A neutral is a commodity unit which does not change utility (gives an equally preferred bundle).
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Goods, Bads and Neutrals
Utility
Water x’
Units of
water are
goods
Units of
water are
bads
Around x’ units, a little extra water is a
neutral.
Utility
function
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Some Other Utility Functions and Their Indifference Curves
• Instead of U(x1,x2) = x1x2 consider V(x1,x2) = x1 + x2. What do the indifference curves for this “perfect substitution” utility function look like?
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Perfect Substitution Indifference Curves
5
5
9
9
13
13
x1
x2
x1 + x2 = 5
x1 + x2 = 9
x1 + x2 = 13
V(x1,x2) = x1 + x2.
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Perfect Substitution Indifference Curves
5
5
9
9
13
13
x1
x2
x1 + x2 = 5
x1 + x2 = 9
x1 + x2 = 13
All are linear and parallel.
V(x1,x2) = x1 + x2.
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Some Other Utility Functions and Their Indifference Curves
• Instead of U(x1,x2) = x1x2 or V(x1,x2) = x1 + x2, consider W(x1,x2) = min{x1,x2}. What do the indifference curves for this “perfect complementarity” utility function look like?
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Perfect Complementarity Indifference Curves
x2
x1
45o
min{x1,x2} = 8
3 5 8
3
5
8
min{x1,x2} = 5
min{x1,x2} = 3
W(x1,x2) = min{x1,x2}
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Perfect Complementarity Indifference Curves
x2
x1
45o
min{x1,x2} = 8
3 5 8
3
5
8
min{x1,x2} = 5
min{x1,x2} = 3
All are right-angled with vertices on a ray
from the origin.
W(x1,x2) = min{x1,x2}
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Some Other Utility Functions and Their Indifference Curves
• A utility function of the form U(x1,x2) = f(x1) + x2 is linear in just x2 and is called quasi-linear.
• E.g. U(x1,x2) = 2x11/2 + x2.
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Quasi-linear Indifference Curves x2
x1
Each curve is a vertically shifted
copy of the others.
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Some Other Utility Functions and Their Indifference Curves
• Any utility function of the form U(x1,x2) = x1
a x2b
with a > 0 and b > 0 is called a Cobb-Douglas utility function.
• E.g. U(x1,x2) = x11/2 x2
1/2 (a = b = 1/2) V(x1,x2) = x1 x2
3 (a = 1, b = 3)
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Cobb-Douglas Indifference Curves x2
x1
All curves are hyperbolic,
asymptoting to, but never
touching any axis.
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Marginal Utilities
• Marginal means “incremental”.
• The marginal utility of commodity i is the rate-of-change of total utility as the quantity of commodity i consumed changes; i.e.
MUU
xi
i
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Marginal Utilities
• E.g. if U(x1,x2) = x11/2 x2
2 then
2
2
2/1
1
1
12
1xx
x
UMU
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Marginal Utilities
• E.g. if U(x1,x2) = x11/2 x2
2 then
2
2
2/1
1
1
12
1xx
x
UMU
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Marginal Utilities
• E.g. if U(x1,x2) = x11/2 x2
2 then
2
2/1
1
2
2 2 xxx
UMU
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Marginal Utilities
• E.g. if U(x1,x2) = x11/2 x2
2 then
2
2/1
1
2
2 2 xxx
UMU
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Marginal Utilities
• So, if U(x1,x2) = x11/2 x2
2 then
2
2/1
1
2
2
2
2
2/1
1
1
1
2
2
1
xxx
UMU
xxx
UMU
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Marginal Utilities and Marginal Rates-of-Substitution
• The general equation for an indifference curve is U(x1,x2) k, a constant. Totally differentiating this identity gives
U
xdx
U
xdx
11
22 0
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Marginal Utilities and Marginal Rates-of-Substitution
U
xdx
U
xdx
11
22 0
U
xdx
U
xdx
22
11
rearranged is
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Marginal Utilities and Marginal Rates-of-Substitution
U
xdx
U
xdx
22
11
rearranged is
And
d x
d x
U x
U x
2
1
1
2
/
/.
This is the MRS.
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Marg. Utilities & Marg. Rates-of-Substitution; An example
• Suppose U(x1,x2) = x1x2. Then
U
xx x
U
xx x
12 2
21 1
1
1
( )( )
( )( )
MRSd x
d x
U x
U x
x
x 2
1
1
2
2
1
/
/.so
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Marg. Utilities & Marg. Rates-of-Substitution; An example
MRSx
x 2
1
MRS(1,8) = - 8/1 = -8
MRS(6,6) = - 6/6 = -1.
x1
x2
8
6
1 6
U = 8
U = 36
U(x1,x2) = x1x2;
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Marg. Rates-of-Substitution for Quasi-linear Utility Functions
• A quasi-linear utility function is of the form U(x1,x2) = f(x1) + x2.
so
U
xf x
11 ( )
U
x2
1
MRSd x
d x
U x
U xf x 2
1
1
21
/
/( ).
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Marg. Rates-of-Substitution for Quasi-linear Utility Functions
• MRS = - f (x1) does not depend upon x2 so the slope of indifference curves for a quasi-linear utility function is constant along any line for which x1 is constant. What does that make the indifference map for a quasi-linear utility function look like?
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Marg. Rates-of-Substitution for Quasi-linear Utility Functions
x2
x1
Each curve is a vertically
shifted copy of the others.
MRS is a
constant
along any line
for which x1 is
constant.
MRS =
- f’(x1’)
MRS = -f’(x1”)
x1’ x1” 71
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Monotonic Transformations & Marginal Rates-of-Substitution
• Applying a monotonic transformation to a utility function representing a preference relation simply creates another utility function representing the same preference relation.
• What happens to marginal rates-of-substitution when a monotonic transformation is applied?
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Monotonic Transformations & Marginal Rates-of-Substitution
• For U(x1,x2) = x1x2 the MRS = - x2/x1.
• Create V = U2; i.e. V(x1,x2) = x12x2
2. What is the MRS for V? which is the same as the MRS for U.
MRSV x
V x
x x
x x
x
x
/
/
1
2
1 22
12
2
2
1
2
2
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Monotonic Transformations & Marginal Rates-of-Substitution
• More generally, if V = f(U) where f is a strictly increasing function, then
MRSV x
V x
f U U x
f U U x
/
/
( ) /
'( ) /
1
2
1
2
U x
U x
/
/.1
2
So MRS is unchanged by a positive
monotonic transformation.
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