西方经济学 微观部分 第四章 效用Ch04Utility
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• 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.
is strictly preferred to bundle y. But x is not preferred three times as much as is y.
p
Utility Functions & Indiff. Curves
• Consider the bundles (4,1), (2,3) and (2,2).
• Another way to visualize this same information is to plot the utility level on a vertical axis.
Utility Functions & Indiff. Curves
3D plot of consumption & utility levels for 3 bundles
• Reflexivity: Any bundle x is always at least as preferred as itself; i.e.
x f~x.
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.
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:
Utility Functions & Indiff. Curves
x2
(2,3) (2,2) ~ (4,1)
p
U6 U4
x1
Utility Functions & Indiff. Curves
as is y.
Preferences - A Reminder
• Completeness: For any two bundles x and
y it is always possible to state either that
xy
or that
y f~x.
f~
Preferences - A Reminder
Utility Functions & Indiff. Curves
• An indifference curve contains equaFra Baidu bibliotekly preferred bundles.
• Equal preference same utility level.
• Therefore, all bundles in an indifference curve have the same utility level.
Utility Functions & Indiff. Curves
Utility
U
U
x2
Higher indifference
curves contain
more preferred
Chapter Four
Utility
p
Preferences - A Reminder
• x y: x is preferred strictly to y. • x ~ y: x and y are equally preferred.
• x f~ y: x is preferred at least as much
• Continuity means that small changes to a consumption bundle cause only small changes to the preference level.
Utility Functions
• A utility function U(x) represents a
preference relation if~f and only if:
x’ x”
U(x’) > U(x”)
p
x’ px”
x’ ~ x”
U(x’) < U(x”) U(x’) = U(x”).
Utility Functions
• Utility is an ordinal (i.e. ordering) concept. • E.g. if U(x) = 6 and U(y) = 2 then bundle x
Utility
U(2,3) = 6
U(2,2) = 4 U(4,1) = 4 x2
x1
Utility Functions & Indiff. Curves
• This 3D visualization of preferences can be made more informative by adding into it the two indifference curves.
x f~y and y f~z
x f~z.
Utility Functions
• A preference relation that is complete, reflexive, transitive and continuous can be represented by a continuous utility function.
• 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.
is strictly preferred to bundle y. But x is not preferred three times as much as is y.
p
Utility Functions & Indiff. Curves
• Consider the bundles (4,1), (2,3) and (2,2).
• Another way to visualize this same information is to plot the utility level on a vertical axis.
Utility Functions & Indiff. Curves
3D plot of consumption & utility levels for 3 bundles
• Reflexivity: Any bundle x is always at least as preferred as itself; i.e.
x f~x.
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.
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:
Utility Functions & Indiff. Curves
x2
(2,3) (2,2) ~ (4,1)
p
U6 U4
x1
Utility Functions & Indiff. Curves
as is y.
Preferences - A Reminder
• Completeness: For any two bundles x and
y it is always possible to state either that
xy
or that
y f~x.
f~
Preferences - A Reminder
Utility Functions & Indiff. Curves
• An indifference curve contains equaFra Baidu bibliotekly preferred bundles.
• Equal preference same utility level.
• Therefore, all bundles in an indifference curve have the same utility level.
Utility Functions & Indiff. Curves
Utility
U
U
x2
Higher indifference
curves contain
more preferred
Chapter Four
Utility
p
Preferences - A Reminder
• x y: x is preferred strictly to y. • x ~ y: x and y are equally preferred.
• x f~ y: x is preferred at least as much
• Continuity means that small changes to a consumption bundle cause only small changes to the preference level.
Utility Functions
• A utility function U(x) represents a
preference relation if~f and only if:
x’ x”
U(x’) > U(x”)
p
x’ px”
x’ ~ x”
U(x’) < U(x”) U(x’) = U(x”).
Utility Functions
• Utility is an ordinal (i.e. ordering) concept. • E.g. if U(x) = 6 and U(y) = 2 then bundle x
Utility
U(2,3) = 6
U(2,2) = 4 U(4,1) = 4 x2
x1
Utility Functions & Indiff. Curves
• This 3D visualization of preferences can be made more informative by adding into it the two indifference curves.
x f~y and y f~z
x f~z.
Utility Functions
• A preference relation that is complete, reflexive, transitive and continuous can be represented by a continuous utility function.