Chapter Six
Demand
Income Changes
A
plot of quantity demanded against
income is called an Engel curve.
Income Changes
x2
Fixed p1 and p2.
y’ < y’’ < y’’’
Income
offer curve
x2’’’
x2’’
x2’
y
y’’’
y’’
y’
x1’ x1’’’
x1’’
x1
Engel
curve;
good 1
x1’ x1’’’ x1*
x1’’
Income Changes and CobbDouglas Preferences
 An
example of computing the
equations of Engel curves; the CobbDouglas case.
a b
U( x1 , x 2 )  x1 x 2 .
 The ordinary demand equations are
*
x1 
ay
by
*
; x2 
.
( a  b)p1
( a  b)p2
Income Changes and CobbDouglas Preferences
*
x1 
ay
by
*
; x2 
.
( a  b)p1
( a  b)p2
Rearranged to isolate y, these are:
( a  b)p1 *
y
x1 Engel curve for good 1
a
( a  b)p2 *
y
x 2 Engel curve for good 2
b
Income Changes and CobbDouglas Preferences
y
y
( a  b)p1 *
y
x1
a
Engel curve
for good 1
x1*
( a  b)p2 *
y
x2
b
x2*
Engel curve
for good 2
Income Changes and PerfectlyComplementary Preferences
 Another
example of computing the
equations of Engel curves; the
perfectly-complementary case.
U( x1 , x 2 )  minx1 , x 2.
 The
ordinary demand equations are
*
*
x1  x 2 
y
.
p1  p2
Income Changes and PerfectlyComplementary Preferences
*
*
x1  x 2 
y
.
p1  p2
Rearranged to isolate y, these are:
*
y  (p1  p2 )x1
*
y  (p1  p2 )x 2
Engel curve for good 1
Engel curve for good 2
Income Changes
Fixed p1 and p2.
x2
y’ < y’’ < y’’’
y
x2’’’
x2’’
x2’
y’’’
y’’
y’
x1’ x1’’’
x1’’
x1
Engel
curve;
good 1
x1’ x1’’’
x1*
x1’’
Quasi-linear Indifference Curves
x2
Each curve is a vertically shifted
copy of the others.
Each curve intersects
both axes.
x1
Income Changes; Quasilinear
Utility
x2
y
~
x1
x1
Engel
curve
for
good 1
x1*
~
x1
Income Changes; Quasilinear
Utility
y
x2
~
x1
x1
Engel
curve
for
good 2
x2*
Income Effects
A
good for which quantity demanded
rises as income increases is called
normal.
 Therefore when a good is normal its
Engel curve must be positively
sloped.
Income Effects
A
good for which quantity demanded
falls as income increases is called
inferior.
 Therefore when a good is income
inferior its Engel curve must be
negatively sloped.
Income Changes; Goods
y
1 & 2 Normal
x2
Income
offer curve
x2’’’
x2’’
x2’
y’’’
y’’
y’
y
y’’’
y’’
y’
x1’ x1’’’
x1’’
x1
Engel
curve;
good 2
x2’ x2’’’
x2’’
x2*
Engel
curve;
good 1
x1’ x1’’’ x1*
x1’’
Income Changes; Good 2 Is Normal,
Good 1 Becomes Income Inferior
x2
Income
offer curve
x1
Income Changes; Good 2 Is Normal,
Good 1 Becomes Income Inferior
x2
m
Engel curve
for good 2
m
x2*
Engel curve
for good 1
x1
x1*
Ordinary Goods
A
good is called ordinary if the
quantity demanded always increases
as its own-price decreases.
Ordinary Goods
Fixed p2 and y.
Downward-sloping
p1 demand curve
x2

p1 price
offer
curve
Good 1 is
ordinary
x 1*
x1
Own-Price Changes
x2
Fixed p2 and y.
p1
Ordinary
demand curve
for commodity 1
p1’’’
p1’’
p1’
x1*(p1’’’)
x1*(p1’)
x1*(p1’’)
x1*(p1’’’)
x1*(p1’)
x1*(p1’’)
x1
x 1*
Own-Price Changes
x2
Fixed p2 and y.
p1 price
offer
curve
p1
Ordinary
demand curve
for commodity 1
p1’’’
p1’’
p1’
x1*(p1’’’)
x1*(p1’)
x1*(p1’’)
x1*(p1’’’)
x1*(p1’)
x1*(p1’’)
x1
x 1*
Own-Price Changes
 The
curve containing all the utilitymaximizing bundles traced out as p1
changes, with p2 and y constant, is
the p1- price offer curve.
 The plot of the x1-coordinate of the
p1- price offer curve against p1 is the
ordinary demand curve for
commodity 1.
Own-Price Changes
 What
does a p1 price-offer curve look
like for a perfect-complements utility
function?
U( x1 , x 2 )  minx1 , x 2.
Then the ordinary demand functions
for commodities 1 and 2 are
Own-Price Changes
*
*
x1 (p1 , p2 , y)  x 2 (p1 , p2 , y) 
y
.
p1  p2
With p2 and y fixed, higher p1 causes
smaller x1* and x2*.
Own-Price Changes
Fixed p2 and y.
p1
p1’’’
x2
p1’’
y/p2
x*2 
Ordinary
demand curve
for commodity 1
is
y
*
x1 
.
p1  p2
p1’
y
p1  p 2
y
p2
x*1 
y
p1  p2
x1
x 1*
Own-Price Changes
 What
does a p1 price-offer curve look
like for a perfect-substitutes utility
function?
U( x1 , x 2 )  x1  x 2 .
Then the ordinary demand functions
for commodities 1 and 2 are
Own-Price Changes
0
*
x1 (p1 , p2 , y)  
, if p1  p2
0
*
x 2 (p1 , p2 , y)  
, if p1  p2
y / p1 , if p1  p2
and
y / p2 , if p1  p2 .
Own-Price Changes
Fixed p2 and y.
p1
Ordinary
demand curve
for commodity 1
p1’’’
y
*
x1 
p1
x2
p2 = p1’’
y
p2
p1 price
offer
curve
p1’


y
*
0  x1 
p2
x1
x 1*
Giffen Goods
 If,
for some values of its own-price,
the quantity demanded of a good
rises as its own-price increases then
the good is called Giffen.
Giffen Goods
Demand curve has
a positively
p1
sloped part
Fixed p2 and y.
x2

p1 price offer
curve
Good 1 is
Giffen
x 1*
x1