L05 Choice Problem: We know – Preferences U ( x 1 , x 2 ) ln x 1 ln x 2 –
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Transcript L05 Choice Problem: We know – Preferences U ( x 1 , x 2 ) ln x 1 ln x 2 –
L05
Choice
Problem:
We know
– Preferences U ( x 1 , x 2 ) ln x 1 ln x 2
– Prices and income p 1 1, p 2 1, m
We want optimal choice
*
1
*
2
(x , x )
10
Secrets of Happiness (SOH):
x2
x1
Abstract approach
In the example we were given
U ( x1 , x 2 ) ln x1 ln x 2
p 1 1, p 2 1, m 10
we found demands - two numbers
x1 5 , x 2 5
Now we use abstract parameters
U ( x1 , x 2 )
p1 , p 2 , m
we find demand functionsNow we
x1 ( p 1 , p 2 , m )
4 types of preferences
x 2 ( p1 , p 2 , m )
Abstract Cobb Douglass Function
Cobb
Douglass utility functions
U ( x1 , x 2 ) x x
a
1
b
2
and
V ( x1 , x 2 ) ln U ( x1 , x 2 )
are equivalent in terms of preferences
Magic (Cobb-Douglass) formula
U ( x1 , x 2 ) a ln x1 b ln x 2
Parameters: a , b , p 1 , p 2 , m
p1 , p 2 , m
Cobb-Douglas: Summary
a b
V
a
ln
x
b
ln
x
U
x
Utility function:
1
2 or
1 x2
Solution:
a m
b m
*
*
x1
, x2
a b p1
a b p2
Shares of income
A) Let U x x
0 .5
1
and p 1 2 , p 2 4 , m 40
0 .5
2
px
, p2 x
x
,x
*
1 1
*
2
*
1
*
2
B) Let U x x
10 20
1
2
and p 1 10 , p 2 10 , m 900
p 1 x1*
, p 2 x 2*
x1*
, x 2*
Interior and corner solution
Interiority
Cobb – Douglass (always interior solution)
a m
x
,
a b p1
*
1
b m
x
a b p2
*
2
SOH: Extreme preferences
Perfect
Complements (shoes)
Perfect
substitutes (cheese)
Perfect Complements: Problem
U ( x1 , x 2 ) min( x1 , x 2 )
p 1 1, p 2 1, m 10
SOH (Perfect Complements)
U ( x1 , x 2 ) min( 2 x1 , x 2 )
p 1 1, p 2 1, m 10
Perfect Complements (SOH)
U ( x1 , x 2 ) min( ax1 , bx 2 )
Interior or corner solution?
p1 , p 2 , m
Is solution always interior?
Not
necessarily
Perfect
Substitutes
Quasilinear
Perfect Substitutes:Problem
U ( x1 , x 2 ) x1 x 2
p 1 1, p 2 2 , m 10
x2
x1
Magic Formula (Substitutes)
U ( x1 , x 2 ) ax1 bx 2
p1 , p 2 , m