7/25/2019 3. Utility

1/4

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

7/25/2019 3. Utility

2/4

complete collection of indi> @urvesreps consumers preferences

collection of all indi>curves for a given preference relation

o indierence #ap

e

7/25/2019 3. Utility

3/4

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

7/25/2019 3. Utility

4/4

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