3. Utility
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Transcript of 3. Utility
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3. UtilityAlgebraic Rep of Preference
Utility Function
Assigns a number to every consumption bundle in a persons preference ordering in accordance wit
2 rules:o If someone is indierent between the bundles assigns number to both bundles
o Has preference of one over the other assigns larger number to the preferred bundle
Continuous utility function:preference relation that is complete, ree!ive, transitive and
continuous
Continuity small changes to consumption bundle creates small changes to preference level
U(x) preference relation only if:
An ordinal"ordering# concept
Example:
$"!# % & ' $"y# % 2 bundle ! is strictly preferred to bundle y
( is )*+ preferred times as much as y
Example: ordinal ranking
-ac.eans publishes uni ran/ings
-c0ill 1better that $*ttawa, $ottawa 1better than 3roc/
4ont have to say *ttawa 56 units better than 3roc/ or -c0ill 5twice as good as 3roc/
Indierence Cur!esExample:
3undles "7,8#"2,#"2,2#
9uppose "2,# "7,8# "2,2#
Assign bundle numbers that preserve preference ordering
o $"2,#%& ; $"7,8#% $"2,2# % 7
o Utility "e!els
Indierence curves contains e
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complete collection of indi> @urvesreps consumers preferences
collection of all indi>curves for a given preference relation
o indierence #ap
e
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E"!8,!2# % minL!8,!2M e!> comps ? operating systems, stereo receivers ?
spea/erso perect complimentsutility function indierence curve
all right angled with vertices on a ray from origin
u"!8,!2# % min La!8,b!2M
+uasi,linear Indierence Cur!e
utility function:
o $"!8,!2# % f"!8# F !2 linear in Nust !2
o !2 % / v"!8# Oheight of each indierence curve, higher /s %
higher indi> @urvesPo u "!8!2# % / % f"!8# F !2
Example:
$"!8,!2# % 2!8C"8K2# F !2
Cobb,-ouglas Indierence Cur!es
$"!8,!2# % !8Ca D !2Cb a ; G ' b ; G
c % aK"a F b#
u"!8,!2# % !8C c D !2 C"8 c#
Example:
$"!8,!2# % !8C"8K2# D !2C"8K2# " a % b % Q#
B"!8,!2# % !8D !2C " a % 8, b % #
lope of Indierence Cur!e: /arginal Utilities
-arginal % incremental marginal utility o commodity i:rateJofJchange of total utility as
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rearranged:
-R9Example:
$"!8,!2# % !8!2 dierentiate each
!8 dierentiated % "8#D !2!2 di % !8D "8#
/R 1 dx2dx4 1 ,x2x4
/R for +uasi,linear Utility Functions
$"!8,!2# % f"!8# F !2o 4ierentiation of f"!8# % f"!8#
o 4ierentiation of !2 % 8
/R% d!2Kd!8 % Jf"!8#
-R9 % J S"!8# doesnt depend on !2
o 9lope % constant along any line for which !8 % constant
Indierence map:
/onotonic *ransfor#ations 0 /R
Apply monotonic to utility functioncreates another utility function representing same preference
relation
+ransforming one set of number into another set preserving order
f"u# rate of change of f"u# as u changes
o L f"u2# f"u8# MK u2 u8
-R9 when monotonic transformation applied
o $"!8,!2# % !8!2 -R9 % J!2K!8
o B % $C2
B"!8,!2# % !8C2 D !2C2
-R9 % J "2!8!2C2#K "2!8C2D!2# % J!2K!8
9ame as $
o 5 1 f(U)
-R9 % J Of"$# ! derive>"$K!8#PK f"$# ! derive>"$K!2#P J d"$K!8# Kd"$K!2#
$nchanged
MRS is unchanged by a positive monotonic transormation