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