, Jan 7th Jan 8 9th
th
Varian and 2025
5 1
.
Optimal Choice 5 . 3 Example
x = () u(y Yz) = U(X Xz)
, ,
Start with u(X x2) ,X u(z 2z)[u(X , xz)
budget : P,X + PeX m
Cobb-douglas
= x ,
·
,
Xz max Xixe St pix + Pete Im
u(0 . 5(6) + (1 -
0 . 5]2) ,
0 .
5(0) + [1-0 574)2u(5
.
.
2) = u(4 2) = u(2 2)
, ,
Pop
,
= X Xz
,
6
slope C d > 0 u= k
m ,
Partial derivatives
↑
X ,
=
1 .
= 2 .
PX , ,
+ PaXz =
m
Next ,
add utility u (X , X2)
= di -
represented by IC u(X Xz) =
,
P(X. )
=
= m
X 2 .
P X . ,
+
optimal utility increases with more
goods (monotonicity) PX + PX , , = m
X=
xt
=
Yo
. ,
,
u= 2 Preference for combination (convex (1 +E )
x , P,
Fi
= m
y = 1
X , + Paxm
X
2 .
P ,
*
(xi)
,
PiX + Put = m.
C+
d Pr
-
Definition: Preferences is monotone if y , X ,
and y2x2 X2 quantity
Optimal point =
income
Implies y > X EP ,
X, + P ,
X, E = m
Demand functions : x, (P ,
P
..
m) or X
=
(P P2 m)
.. .
Definition : Preferences is convex if yex and 22x X . (i) =
Implies dy + (1-0] 2 *X for d in 10 1) , X
, = PerfectCompliments X= = X ,
u(X Xz) .
= min EX Xa ,
Y =2
Optimal Choice is where to
budget line
IC is
tangent 1 X .
,
= X2 V= 1
Slope of
budget line is X =P +
X
is
,
Slope of IL
·
2 . P , X, +
P2X = m
P ,X ,
+ PaX ,
= m
the
M
implicit function thea X,
-
p +P
=
,
Perfect Substitutes UNE
Optimal Choice is d
at u(x Xz) X
,
=
,
+ xz
of
utility
value
XP. Pe
- P.
P
is
xi
Definition: MRS k
=-
-
x==
twice a ↑ X
du Popu
as mc
Say
= 2- likes good
-slope Up
slope-
-
-
Say P = - buy good a
X
E
O if P.P ,
x m .
-
PX i
if p , = P2
m 7RP
,
=
e = +x b)
(n - ( (b E
=
-
=
o = 10) 1 J
=
= ③
-
-
=
-X +
(n x(b +y0)4 y 2
Y
+ 4 =y
~
Y
-y +y = (44) &
-Y
-
o = -
(a +x+a + 140) sendso
-
Yy + (x(4 =
(24-4)-
e=
-
=
y y 7 Senatend seaw/1211D
-
* 90mmx12 9
-
o a)
-
G -
J=
.
(n +x20 + y b)4 (4) 44 + 14(41) = 1
=
(4(4(4 + 44(417 (24(X) &
e
= =
44 4)2 - =
(44 (4734j4
o = + exw42n 404 350) .
-= +
2801010 + 01 & &nes i
·
Audio
((4 4/4/72(74(X(4)74847 (44/42(44)4 des
#a -
- =
(n - y0 + ya)4 (44(4 (4 XX)1 v enendeyin0 414484 . .
10744es ve basemen 14 :
114)(4)+ 30/05)4)X 104440)
ebuendey 2019
dursuant waterint do pothmeno to doory wee send
sacpye anns open chenle (14141417 104411441240174411(144111 44) .
:+ 20 Jenna) XP10004314404)
panphemoun evi volgend payment nn 2201 wid we
th
Varian and 2025
5 1
.
Optimal Choice 5 . 3 Example
x = () u(y Yz) = U(X Xz)
, ,
Start with u(X x2) ,X u(z 2z)[u(X , xz)
budget : P,X + PeX m
Cobb-douglas
= x ,
·
,
Xz max Xixe St pix + Pete Im
u(0 . 5(6) + (1 -
0 . 5]2) ,
0 .
5(0) + [1-0 574)2u(5
.
.
2) = u(4 2) = u(2 2)
, ,
Pop
,
= X Xz
,
6
slope C d > 0 u= k
m ,
Partial derivatives
↑
X ,
=
1 .
= 2 .
PX , ,
+ PaXz =
m
Next ,
add utility u (X , X2)
= di -
represented by IC u(X Xz) =
,
P(X. )
=
= m
X 2 .
P X . ,
+
optimal utility increases with more
goods (monotonicity) PX + PX , , = m
X=
xt
=
Yo
. ,
,
u= 2 Preference for combination (convex (1 +E )
x , P,
Fi
= m
y = 1
X , + Paxm
X
2 .
P ,
*
(xi)
,
PiX + Put = m.
C+
d Pr
-
Definition: Preferences is monotone if y , X ,
and y2x2 X2 quantity
Optimal point =
income
Implies y > X EP ,
X, + P ,
X, E = m
Demand functions : x, (P ,
P
..
m) or X
=
(P P2 m)
.. .
Definition : Preferences is convex if yex and 22x X . (i) =
Implies dy + (1-0] 2 *X for d in 10 1) , X
, = PerfectCompliments X= = X ,
u(X Xz) .
= min EX Xa ,
Y =2
Optimal Choice is where to
budget line
IC is
tangent 1 X .
,
= X2 V= 1
Slope of
budget line is X =P +
X
is
,
Slope of IL
·
2 . P , X, +
P2X = m
P ,X ,
+ PaX ,
= m
the
M
implicit function thea X,
-
p +P
=
,
Perfect Substitutes UNE
Optimal Choice is d
at u(x Xz) X
,
=
,
+ xz
of
utility
value
XP. Pe
- P.
P
is
xi
Definition: MRS k
=-
-
x==
twice a ↑ X
du Popu
as mc
Say
= 2- likes good
-slope Up
slope-
-
-
Say P = - buy good a
X
E
O if P.P ,
x m .
-
PX i
if p , = P2
m 7RP
,
=
e = +x b)
(n - ( (b E
=
-
=
o = 10) 1 J
=
= ③
-
-
=
-X +
(n x(b +y0)4 y 2
Y
+ 4 =y
~
Y
-y +y = (44) &
-Y
-
o = -
(a +x+a + 140) sendso
-
Yy + (x(4 =
(24-4)-
e=
-
=
y y 7 Senatend seaw/1211D
-
* 90mmx12 9
-
o a)
-
G -
J=
.
(n +x20 + y b)4 (4) 44 + 14(41) = 1
=
(4(4(4 + 44(417 (24(X) &
e
= =
44 4)2 - =
(44 (4734j4
o = + exw42n 404 350) .
-= +
2801010 + 01 & &nes i
·
Audio
((4 4/4/72(74(X(4)74847 (44/42(44)4 des
#a -
- =
(n - y0 + ya)4 (44(4 (4 XX)1 v enendeyin0 414484 . .
10744es ve basemen 14 :
114)(4)+ 30/05)4)X 104440)
ebuendey 2019
dursuant waterint do pothmeno to doory wee send
sacpye anns open chenle (14141417 104411441240174411(144111 44) .
:+ 20 Jenna) XP10004314404)
panphemoun evi volgend payment nn 2201 wid we
