L06
Demand
Review


Model of choice U  x1 x 2
parameters p1 , p 2 , m
x1 ( p 1 , p 2 , m )

x 2 ( p1 , p 2 , m )
Example 1: Cobb Douglass
U ( x1 , x 2 )  x x 
a b
1 2
Perfect Complements
U  min( ax1 , bx2 ) p1 , p 2 , m
Perfect Substitutes:Problem
U ( x1 , x 2 )  x1  x 2
p 1  1, p 2  2 , m  1 0
x2
x1
Magic Formula (Substitutes)
U ( x1 , x 2 )  a x1  b x 2
p1 , p 2 , m
Comparative statics
*
1
*
2

We know x ( p 1 , p 2 , m ),

Focus on one good (x1)

How the demand is affected by a change
a) in “own” price
b) in income
c) in price of other commodity

One variable at the time!
x ( p1 , p 2 , m )
Own-Price Changes
 We
 We
focus on good 1
*
x1 ( p 1 , p 2 , m )
hold p2 and m constant.
 We change p1
 The change represented by:
- Price offer curve
- Demand curve
Own-Price Change p1
Fix p2=1 and m=12.
x2
(2.5,3)
Vary p1=1, p1’=3, p1’’=4
p1 price offer
curve
p1
Demand curve
for commodity 1
(5,7)
(3,3)
x1
x 1*
Own-Price Changes
 The
curve containing all the utilitymaximizing bundles traced out as p1
changes, with p2 and m constant, is
the p1- price offer curve.
 The plot of optimal choice of x1
against p1 is the demand curve for
commodity 1.
Ordinary and Giffen goods
p1
x 1*
Cobb-Douglas example
We find price offer and demand curve for
 Cobb-Douglas preferences
U ( x1 , x 2 )  x x .
2
1

We keep fixed
p 2  1, m  1 2
2
2
Cobb-Douglass example
Data U  x x ,
p 2  1, m  12 , variable p1
x ( p1 , p 2 , m ) 
x ( p1 , p 2 , m ) 
2 2
1 2
*
1
*
2
Quiz
For Cobb-Douglass
A) Price offer curve flat
B) Demand downwar-slopping
Q1: For Cobb-Douglas preferences commodities
A) are ordinary goods
B) are Giffen goods
C) Depends on the parameters
D) I do not know
Income Changes
 We
 We
still focus on good 1
*
x1 ( p 1 , p 2 , m )
hold p1 and p2 constant.
 We change m
 The change represented by:
- Income offer curve
- Engel curve
Income Changes
Fix p1=1, p2=1
x2
Vary m=12, m’=6, m’’=4
income offer
curve
m
Engel curve
for commodity 1
(5,7)
(3,3)
(2,2)
x1
x 1*
Goods
A
good for which quantity demanded
rises with income is called normal.
(positive slope of Engel curve)
 A good for which quantity demanded
falls as income increases is called
income inferior.
(negative slope of Engel curve)
Cobb-Douglas example
We find income offer and Engel curve for
 Cobb-Douglas preferences
U ( x1 , x 2 )  x x .
2
1

In both cases we assume
p 1  1, p 2  1
2
2
Cobb-Douglass example
Data U  x x ,
2 2
1 2
x ( p1 , p 2 , m ) 
*
1
p 1  1, p 2  1
, variable
x ( p1 , p 2 , m ) 
*
2
m
Quiz
For Cobb-Douglass
A) Income offer curve- ray from origin
B) Engel curve upward-slopping
Q1: For Cobb-Douglas preferences commodities
A) are normal goods
B) are inferior goods
C) Depends on the parameters
D) I do not know
Cross-Price Effects
 If
an increase in p2
– increases demand for commodity 1
then commodity 1 is a gross
substitute for commodity 2.
– reduces demand for commodity 1
then commodity 1 is a gross
complement for commodity 2.
Cobb Douglas example
Gross complements of substitutes?
Perfect Complements example
Gross complements