Over view lecture 1
Axioms
1
Complete Ness
It 't
'
' ' '
t × × E X either X or × ×
,
utility function
,
.
2
Transitiviteit A function U :
IRI → IR is called a
' ' '
't 't
' '
For × , × × EX If × × × ×
any ,
utility the
,
represented
,
function
'
then x 't ×
relation if
.
prefereren 2
3 continuit '
) xo EIRI
'
ulx
'
) ± ulx
'
⇐ Ex t xo, ×
For an X C- IRI ,
L (x ) and I (× ) are
- invariante to positive monotonie
closed in IRI .
transformaties .
4 Local non -
satiation
→ IK ) =
{ × 1×2×03=9×1 ulx ) zulk ) }
For an I C- IRI and HE > 0
,
] some
is called a
superiors set .
'
X E B , (xo ) such that × > × .
Quasi concavity
5 Strict monotone city "
let f :D → IR ,
where DE IR and convex .
ER ?
'
xozx
'
xo It
'
t
,
x :
.
if ,
then ×
' '
* Quasiaancare iff t x
,
× E D
xo >
'
It
'
.
if ×
,
then ×
text ) { f- (x ) f- LI ) }
'
= min
6
,
Convexity '
with xt =
tx
'
+ ( 1- f) x t te [0,13
XIX EIRI
' ' '
t
,
it X 2X ,
then
* Quasi c o nvex
( 1- f) Ik
'
[ 0,13
'
tx 1- × t te
f- txt ) { f- (× ) f- ( x2 ) }
'
E Max ,
7 Strict convexity
' '
IRI
'
t xo # X E If × 2x ,
then
,
)
' '
1- × t ( i -
E) xo ×
× t te ( 0,1
S. Veeling
ij
, Over view lecture 2
Utility maximization
problem
.
Budget set B =
Ex ) XE IRI , D. × Ey } Marginet rate of substituties MRS
Max ulx ) ⇐ (x ) * MRS =
d at =
#
in
u
B die Itxz P2
sit .
p
-
×
Eg
is the optimale t condition
This can be explained by the
following
Calculatie
:
solution
*
optimal
f. ( × 7) =
ulx ) X ( p )
→
ya
se " Eso units of goed 1
,
which
×
y
-
-
-
,
deoeasesyovr utility by I c .
* KKT conditions
pas
-
:
JX ,
- om × :
Xi 20 Xi I =
0 I to with the extra
, , → nou
you buygood 2
di di
income units
E-
p ,
,
so
you buy E.pe
t
.
→ om 2 A zo X 0 I to
pz
: =
, ,
JX .
utility by
¥-2 Eppi
This i n c re a s e
-
*
17 ulx ) Strictly quasicomcare , then UMP
→ In optimal situation those two are equal .
has a
Unique Solution .
0¥
Ja EPI = < ⇐ MRS
* Solution to UMP Maxlp ) is called I §
, , y
the Marshallion demand function .
S. Veeling
ij
Axioms
1
Complete Ness
It 't
'
' ' '
t × × E X either X or × ×
,
utility function
,
.
2
Transitiviteit A function U :
IRI → IR is called a
' ' '
't 't
' '
For × , × × EX If × × × ×
any ,
utility the
,
represented
,
function
'
then x 't ×
relation if
.
prefereren 2
3 continuit '
) xo EIRI
'
ulx
'
) ± ulx
'
⇐ Ex t xo, ×
For an X C- IRI ,
L (x ) and I (× ) are
- invariante to positive monotonie
closed in IRI .
transformaties .
4 Local non -
satiation
→ IK ) =
{ × 1×2×03=9×1 ulx ) zulk ) }
For an I C- IRI and HE > 0
,
] some
is called a
superiors set .
'
X E B , (xo ) such that × > × .
Quasi concavity
5 Strict monotone city "
let f :D → IR ,
where DE IR and convex .
ER ?
'
xozx
'
xo It
'
t
,
x :
.
if ,
then ×
' '
* Quasiaancare iff t x
,
× E D
xo >
'
It
'
.
if ×
,
then ×
text ) { f- (x ) f- LI ) }
'
= min
6
,
Convexity '
with xt =
tx
'
+ ( 1- f) x t te [0,13
XIX EIRI
' ' '
t
,
it X 2X ,
then
* Quasi c o nvex
( 1- f) Ik
'
[ 0,13
'
tx 1- × t te
f- txt ) { f- (× ) f- ( x2 ) }
'
E Max ,
7 Strict convexity
' '
IRI
'
t xo # X E If × 2x ,
then
,
)
' '
1- × t ( i -
E) xo ×
× t te ( 0,1
S. Veeling
ij
, Over view lecture 2
Utility maximization
problem
.
Budget set B =
Ex ) XE IRI , D. × Ey } Marginet rate of substituties MRS
Max ulx ) ⇐ (x ) * MRS =
d at =
#
in
u
B die Itxz P2
sit .
p
-
×
Eg
is the optimale t condition
This can be explained by the
following
Calculatie
:
solution
*
optimal
f. ( × 7) =
ulx ) X ( p )
→
ya
se " Eso units of goed 1
,
which
×
y
-
-
-
,
deoeasesyovr utility by I c .
* KKT conditions
pas
-
:
JX ,
- om × :
Xi 20 Xi I =
0 I to with the extra
, , → nou
you buygood 2
di di
income units
E-
p ,
,
so
you buy E.pe
t
.
→ om 2 A zo X 0 I to
pz
: =
, ,
JX .
utility by
¥-2 Eppi
This i n c re a s e
-
*
17 ulx ) Strictly quasicomcare , then UMP
→ In optimal situation those two are equal .
has a
Unique Solution .
0¥
Ja EPI = < ⇐ MRS
* Solution to UMP Maxlp ) is called I §
, , y
the Marshallion demand function .
S. Veeling
ij
