L05
Choice
Problem:

We know preferences (utility function)
U ( x1 , x 2 )  ln x1  ln x 2
and

p1  1, p 2  1, m  10
We want to know optimal choice
*
1
*
2
(x , x )
Choice
U ( x1 , x 2 )  ln x1  ln x 2
p1  1, p 2  2, m  10
MU 1
MU 2
p1
p2
$
$
$
$
$
$
$
$
$
$
Choice: geometric solution
x2
x1
Abstract approach
In the example we were given
p1  1, p 2  1, m  10
U ( x1 , x 2 )  ln x1  ln x 2
we found demands - two numbers
x1  5, x 2  5
 Now we use abstract parameters
p1 , p 2 , m
U ( x1 , x2 )
we find demand functionsNow we

x1 ( p1 , p 2 , m )
4 types of preferences
x2 ( 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
Abstract Cobb Douglass Function
U ( x1 , x 2 )  x x
a
1
b
2
V ( x1 , x 2 )  a ln x1  b ln x 2
Magic (Cobb-Douglass) formula
U ( x1 , x 2 )  a ln x1  b ln x 2
Parameters: a , b , p1 , p 2 , m
p1 , p 2 , m
Cobb-Douglas: Summary
a b
V

a
ln
x

b
ln
x
U

x
x
Utility function:
1
2 or
1 2
Solution:
x 
*
1
a
m
a  b p1
Shares of income
,
x 
*
2
b
m
a  b p2
A) Let U  x x
0 .5
1
and p1  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
1
and p1  10, p 2  10, m  900
20
2
p1 x1 
, p2 x2 
x1 
, x2 
*
*
*
*
Interiority
Cobb – Douglass (always interior solution)
MU 1
lim
MU 1
p1
xi 0
MU 2
p2
Cobb- Douglass preferences
x2
x1
SOH (Perfect Complements)
U ( x1 , x 2 )  min( x1 , x 2 )
p1  1, p 2  1, m  10
SOH (Perfect Complements)
U ( x1 , x2 )  min( 2 x1 , x2 )
p1  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
 Even with well behaved preferences
we might have a corner solution
 Example:
Perfect Substitutes
Perfect substitutes
U ( x1 , x 2 )  x1  x 2
p1  1, p 2  2, m  10
MU 1
MU 2
p1
p2
$
$
$
$
$
$
$
$
$
$
Perfect Substitutes
U ( x1 , x 2 )  x1  x 2
p1  1, p 2  2, m  10
x2
x1
Magic (Substitutes) Formula
U ( x1 , x 2 )  ax1  bx 2
p1 , p 2 , m
Choice
U ( x1 , x 2 )  x1  20 ln x 2
p1  1, p 2  1, m  10
MU 1
MU 2
p1
p2
$
$
$
$
$
$
$
$
$
$
Is solution interior?

Hence demand x  0
*
1
MU 1
p1


1)
2)

MU 2
p2

and x 
MU 1
MU 2
*
2
m
p2
| MRS |
p1
p2
Geometric interpretation
How to solve for corner solution?
Find a buddle using standard conditions
If some xi  0 then in optimum xi*  0
Choice: Calculation
U ( x1 , x 2 )  x1  20 ln x 2
p1  1, p 2  1, m  10
In Practice

Cobb-Douglass, Perfect Complements?

Quasilinear ?

Perfect Substitutes?