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Economics 30 30: Intermediate Microeconomic Theory
Topic 2: Consumer Theory
Cobb-Douglas Preferences
Reference: Varian, p.
Outline: 1. Introduction
_2. Cobb-Douglass Preferences
- Positive Monotonic Transformations
- Marginal Rate of Substitution_
1. Introduction
Cobb-Douglass preferences are one of the simplest algebraic representations of well-behaved
preferences.
2. Cobb-Douglas Preferences
Assume the consumer’s utility function is given by:
u x 1
, x 2 ( ) =^ x 1
c x 1
d (1)
We will maximise this utility function subject to the following budget constraint:
p 1
x 1
x 2
= m (2)
Thus, setting up the Lagrange multiplier:
1
max
x 1 , x 2 , λ
L = x 1
c x 2
d
x 1
− p 2
x 2 ( ) (3)
∂ L
∂ x 1
= cx 1
c − 1 x 2
d − λ p 1
∂ L
∂ x 2
= dx 1
c x 2
d − 1 − λ p 2
∂ L
∂ λ
= m − p 1
x 1
− p 2
x 2
Dividing (4) by (5):
1 See the Appendix to Chapter 4 of Varian for details.
cx 1
c − 1 x 2
d
dx 1
c x 2
d − 1
p 1
p 2
c
d
x 1
c − 1 − c x 2
d − d + 1
p 1
p 2
c
d
x 2
x 1
p 1
p 2
x 1
c
d
p 2
p 1
⋅ x 2
Also note from (6) that:
x 2
m
p 2
p 1
p 2
x 1
We now have two equations - (7) and (8) – and two unknowns x 1
, x 2 ( ). Substituting (8) into (7)
implies:
x 1
c
d
p 2
p 1
x 2
x 1
c
d
p 2
p 1
m
p 2
p 1
p 2
x 1
x 1
c
d
m
p 1
c
d
⋅ x 1
x 1
c
d
c
d
m
p 1
x 1
d
c + d
c
d
m
p 1
x 1
c
c + d
m
p 1
p 1
x 1
m
p 2
x 2
m
c
c + d
d
c + d
3. Positive Monotonic Transformation
Recall equation (1):
u x 1
, x 2
( ) =^ x
1
c x 1
d (14)
If we make a positive monotonic transformation of the Cobb-Douglas utility function by raising
it to the 1 ( c + d ) power, we derive:
v x 1
, x 2
( ) =^ u^ x
1
, x 2
1
c + d = x 1
c x 1
d
1 c + d = x 1
c c + d x 1
d c + d (15)
Define a new number a = c ( c + d ) enables us to rewrite (15) as:
v x 1
, x 2
( ) = x
1
c c + d x 1
d c + d = x 1
a x 1
1 − a (16)
Thus, we can always take a positive monotonic transformation of Cobb-Douglass preferences
that will make the exponents sum to 1. This is particularly useful, because it implies:
x 1
c
c + d
m
p 1
= a
m
p 1
x 2
d
c + d
m
p 2
= ( 1 − a )
m
p 2
Such that the consumer’s expenditure shares are given by:
p 1
x 1
m
= a
p 2
x 2
m
= 1 − a
Another useful positive monotonic transformation of the Cobb- Douglas utility is to take logs vis :
w x 1
, x 2
( ) =^ ln^ u^ x
1
, x 2
( ) =^ ln^ x
1
c x 1
d
( ) =^ ln^ x
1
c
( ) +^ ln^ x
1
d
( ) =^ c^ ln^ x
1
4. Marginal Rate of Substitution
Generally, if u = u x 1
, x 2
( ) , then:
du x 1
, x 2 ( ) =^ u x 1
dx 1
dx 2
dx 2
dx 1 MRS
u x 1
u x 2
d
2 x 2
dx 1
2
MRS
u x 1 x 1
u x 2
From (19):
dx 2
dx 1 MRS
u x 1
u x 2
c x 1
d x 2
cx 2
dx 1
Equivalently, from (1):
dx 2
dx 1 MRS
u x 1
u x 2
cx 1
c − 1 x 2
d
dx 1
c x 2
d − 1
cx 1
c − 1 − c
dx 2
d − 1 − d
cx 2
dx 1
