Econ 400B Lecture Note.pdf - Chapter 3 Preferences and...

Chapter 3: Preferences and Utility
° There are n commodities (also include time and location). A consumption bundle
is denoted as X = ( x
1
; x
2
; : : : ; x
n
) 2 R n
+
:
° A consumer °should be able±to make simple choice between two consumption bundles,
known as preference relation .
° If the consumer likes bundle A better than bundle B , then ° A is (strictly) preferred
to B ±, denote as A ± B .
° If the consumer thinks that A and B are equally attractive, then ° A is indi/erent
from B ±, denote as A ² B .
° If either A ± B or A ² B , then ° A is weakly preferred to B ±, or A is at least as
good as B , denote as A % B:
° Axioms of Rational Choice
° Complete: For all A and B , either A ± B , or B ± A , or A ² B .
° Transitive: If A ± B and B ± C , then A ± C .
° Continuous: If A ± B , and C is °very close to± A , then C ± B:
Lexicographic preference is not continuous!
° U ( ³ ) : R n ! R is a utility function if
U ( A ) > U ( B ) if and only A ± B , and U ( A ) = U ( B ) if and only A ² B .
° If there is a utility function, then preference must be complete and transitive.
° If preference is complete, transitive, and continuous , then there is a utility function.
° Any monotonic transformation of a utility function is also a utility function.
° With n commodities, denote a bundle as A = ( x
1
; x
2
; : : : ; x
n
) , and utility as
U ( A ) = U ( x
1
; x
2
; : : : ; x
n
) :
With two commodities ( n = 2 ), simply write a utility function as U ( x; y ) .
° An indi/erence curve is a set of all bundles that are indi/erent from each other.
° All bundles on the same indi/erence curve have the same utility.
° Given utility function U ( x; y ) , an indi/erence curve is f ( x; y ) : U ( x; y ) = ° g ,
where ° is the °common utility±of the bundles on the indi/erence curve.
° How to ²nd an indi/erence curve?
Set U ( x; y ) = ° , then solve y as a function of x (and ° ).
° Marginal Rate of Substitution ( MRS )
° The slope of an indi/erence curve (at a point/bundle) MRS = ´
dy
dx
°
°
U = °
:
1

° What is the meaning of MRS ? The consumer is willing to exchange one unit
of good X with MRS units of good Y .
° Indi/erence curve map
° Strictly and weakly convex preference
° A convex combination of two bundles ( x
1
; y
1
) and ( x
2
; y
2
) is, for ± 2 (0 ; 1) ,
± ( x
1
; y
1
) + (1 ´ ± )( x
2
; y
2
) = ( ±x
1
+ (1 ´ ± ) x
2
; ±y
1
+ (1 ´ ± ) y
2
)
° Preference µ is strictly convex if for ( x
1
; y
1
) ² ( x
2
; y
2
) and ± 2 (0 ; 1) ,
± ( x
1
; y
1
) + (1 ´ ± )( x
2
; y
2
) ± ( x
1
; y
1
) ² ( x
2
; y
2
) :
Preference µ is weakly convex if for ( x
1
; y
1
) ² ( x
2
; y
2
) and ± 2 (0 ; 1) ,
± ( x
1
; y
1
) + (1 ´ ± )( x
2
; y
2
) µ ( x
1
; y
1
) ² ( x
2
; y
2
) :
-
6
x
y
0
r
U
1
@
@
@
@
@
@
r
r
r
( x
2
; y
2
)
( x
C
; y
C
)
( x
1
; y
1
)
convex
-
6
x
y
0
r
U
1
@
@
@
@
@
@
r
r
r
( x
2
; y
2
)
( x
C
; y
C
)
( x
1
; y
1
)
nonconvex
° Example 3.1 Utility function U ( x; y ) =
p
xy
° To ²nd the indi/erence curve with utility 10 ;
p
xy = 10 ) y =
100
x
° To ²nd marginal rate of substitution: MRS = ´
dy
dx
=
100
x
2
° To demonstrate convexity:
Note that (20 ; 5) ² (5 ; 20) as U (20 ; 5) = U (5 ; 20) = 10 :
U
±
1
2
(20 ; 5) + 1
2
(5 ; 20)
²
= U (12 : 5 ; 12 ; 5) =
p
12 : 5 ¶ 12 : 5 = 12 : 5 > 10 .
° Marginal utilities: Additional utility from consuming more but small amount of one
commodity
MU
x
=
@U ( x; y )
@x
and MU
y
=
@U ( x; y )
@y
:
° How to ²nd marginal utility? Two ways: derivative of an indi/erence curve, or
° Marginal rate of substitution and marginal utility: MRS = ´
dy
dx
°
°
U ( x;y )= °
=
MU
x
MU
y
Note that U ( x; y ) = ° , then
@U ( x;y )
@x
dx +
@U ( x;y )
@y
dy = 0 ) MRS = ´
dy
dx
=
MU
x
MU
y
:
2

°