Summary pareto efficiency, econ 391

, 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) .
-= +
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