Chapter Six
Demand
Properties of Demand Functions
Comparative statics analysis （ 比较静态分析） of ordinary demand
functions -- the study of how
ordinary demands x1*(p1,p2,y) and
x2*(p1,p2,y) change as prices p1,p2
and income y change.
Structure
Own-price changes
– Price offer curve （ 价格提供曲线）
– Ordinary demand curve
– Inverse demand curve （ 反需求函数）
Income changes
– Income offer curve （ 收入提供曲线）
– Engel curve （ 恩格尔曲线）
Cross-price effects
Own-Price Changes
How does x1*(p1,p2,y) change as p1
changes,holding p2 and y constant?
Suppose only p1 increases,from p1’
to p1’’ and then to p1’’’.
x1
x2
p1 = p1’
Fixed p2 and y.
p1x1 + p2x2 = y
Own-Price Changes
Own-Price Changes
x1
x2
p1= p1’’
p1 = p1’
Fixed p2 and y.
p1x1 + p2x2 = y
Own-Price Changes
x1
x2
p1= p1’’
p1=
p1’’’
Fixed p2 and y.
p1 = p1’
p1x1 + p2x2 = y
x
2
x
1
p1 = p1’
Own-Price Changes
Fixed p2 and y.
x
2
x
1
x1*(p1’)
Own-Price Changes
p1 = p1’
Fixed p2 and y.
x
2
x
1
x1*(p1’)
p1
x1*(p1’)
p1’
x1*
Own-Price Changes
Fixed p2 and y.
p1 = p1’
x
2
x
1
x1*(p1’)
p1
x1*(p1’)
p1’
p1 = p1’’
x1*
Own-Price Changes
Fixed p2 and y.
x
2
x
1
x1*(p1’)
x1*(p1’’)
p1
x1*(p1’)
p1’
p1 = p1’’
x1*
Own-Price Changes
Fixed p2 and y.
x
2
x
1
x1*(p1’)
x1*(p1’’)
p1
x1*(p1’)
x1*(p1’’)
p1’
p1’’
x1*
Own-Price Changes
Fixed p2 and y.
x
2
x
1
x1*(p1’)
x1*(p1’’)
p1
x1*(p1’)
x1*(p1’’)
p1’
p1’’
p1 = p1’’’
x1*
Own-Price Changes
Fixed p2 and y.
x
2
x
1
x1*(p1’’’) x1*(p1’)
x1*(p1’’)
p1
x1*(p1’)
x1*(p1’’)
p1’
p1’’
p1 = p1’’’
x1*
Own-Price Changes
Fixed p2 and y.
x
2
x
1
x1*(p1’’’) x1*(p1’)
x1*(p1’’)
p1
x1*(p1’)x1*(p1’’’)
x1*(p1’’)
p1’
p1’’
p1’’’
x1*
Own-Price Changes
Fixed p2 and y.
x
2
x
1
x1*(p1’’’) x1*(p1’)
x1*(p1’’)
p1
x1*(p1’)x1*(p1’’’)
x1*(p1’’)
p1’
p1’’
p1’’’
x1*
Own-Price Changes Ordinarydemand curve
for commodity 1Fixed p2 and y.
x
2
x
1
x1*(p1’’’) x1*(p1’)
x1*(p1’’)
p1
x1*(p1’)x1*(p1’’’)
x1*(p1’’)
p1’
p1’’
p1’’’
x1*
Own-Price Changes Ordinarydemand curve
for commodity 1Fixed p2 and y.
x
2
x
1
x1*(p1’’’) x1*(p1’)
x1*(p1’’)
p1
x1*(p1’)x1*(p1’’’)
x1*(p1’’)
p1’
p1’’
p1’’’
x1*
Own-Price Changes Ordinarydemand curve
for commodity 1
p1 price
offer
curve
Fixed p2 and y.
Own-Price Changes
The curve containing all the utility-
maximizing 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 Cobb-Douglas preferences?
Own-Price Changes
What does a p1 price-offer curve look
like for Cobb-Douglas preferences?
Take
Then the ordinary demand functions
for commodities 1 and 2 are
U x x x xa b(,),1 2 1 2?
Own-Price Changes
x p p y a
a b
y
p1 1 2 1
* (,,)?
x p p y b
a b
y
p2 1 2 2
* (,,),?
and
Notice that x2* does not vary with p1 so the
p1 price offer curve is
Own-Price Changes
x p p y a
a b
y
p1 1 2 1
* (,,)?
x p p y b
a b
y
p2 1 2 2
* (,,),?
and
Notice that x2* does not vary with p1 so the
p1 price offer curve is flat
Own-Price Changes
x p p y a
a b
y
p1 1 2 1
* (,,)?
x p p y b
a b
y
p2 1 2 2
* (,,),?
and
Notice that x2* does not vary with p1 so the
p1 price offer curve is flat and the ordinary
demand curve for commodity 1 is a
Own-Price Changes
x p p y a
a b
y
p1 1 2 1
* (,,)?
x p p y b
a b
y
p2 1 2 2
* (,,),?
and
Notice that x2* does not vary with p1 so the
p1 price offer curve is flat and the ordinary
demand curve for commodity 1 is a
rectangular hyperbola.
x1*(p1’’’) x1*(p1’)
x1*(p1’’)
x
2
x
1
Own-Price Changes
Fixed p2 and y.x
by
a b p
2
2
*
( )
x aya b p1
1
*
( )
x1*(p1’’’) x1*(p1’)
x1*(p1’’)
x
2
x
1
p1
x1*
Own-Price Changes Ordinarydemand curve
for commodity 1
is
Fixed p2 and y.x
by
a b p
2
2
*
( )
x aya b p1
1
*
( )
x aya b p1
1
*
( )
Own-Price Changes
What does a p1 price-offer curve look
like for a perfect-complements utility
function?
Own-Price Changes
What does a p1 price-offer curve look
like for a perfect-complements utility
function?
U x x x x(,) m i n,.1 2 1 2?
Then the ordinary demand functions
for commodities 1 and 2 are
Own-Price Changes
x p p y x p p y y
p p1 1 2 2 1 2 1 2
* *(,,) (,,),
Own-Price Changes
x p p y x p p y y
p p1 1 2 2 1 2 1 2
* *(,,) (,,),
With p2 and y fixed,higher p1 causes
smaller x1* and x2*.
Own-Price Changes
x p p y x p p y y
p p1 1 2 2 1 2 1 2
* *(,,) (,,),
With p2 and y fixed,higher p1 causes
smaller x1* and x2*.
p x x y
p1 1 2 2
0,.* *As
Own-Price Changes
x p p y x p p y y
p p1 1 2 2 1 2 1 2
* *(,,) (,,),
With p2 and y fixed,higher p1 causes
smaller x1* and x2*.
p x x y
p1 1 2 2
0,.* *As
p x x1 1 2 0,.* *As
Fixed p2 and y.
Own-Price Changes
x1
x2
p1
x1*
Fixed p2 and y.x
y
p p
2
1 2
*?
x yp p1
1 2
*?
Own-Price Changes
x1
x2
p1’
x yp p1
1 2
*?
’
p1 = p1’
’
’
y/p2
p1
x1*
Fixed p2 and y.x
y
p p
2
1 2
*?
x yp p1
1 2
*?
Own-Price Changes
x1
x2
p1’
p1’’
p1 = p1’’
’’
x yp p1
1 2
*?
’’
’’
y/p2
p1
x1*
Fixed p2 and y.
x
y
p p
2
1 2
*?
x yp p1
1 2
*?
Own-Price Changes
x1
x2
p1’
p1’’
p1’’’
x yp p1
1 2
*?
p1 = p1’’’
’’’
’’’
’’’
y/p2
p1
x1*
Ordinary
demand curve
for commodity 1
is
Fixed p2 and y.x
y
p p
2
1 2
*?
x yp p1
1 2
*?
x yp p1
1 2
*,?
Own-Price Changes
x1
x2
p1’
p1’’
p1’’’
y
p2
y/p2
Own-Price Changes
What does a p1 price-offer curve look
like for a perfect-substitutes utility
function?
U x x x x(,),1 2 1 2
Then the ordinary demand functions
for commodities 1 and 2 are
Own-Price Changes
x p p y
if p p
y p if p p1 1 2
1 2
1 1 2
0*
(,,)
,
/,
x p p y
if p p
y p if p p2 1 2
1 2
2 1 2
0*
(,,)
,
/,.
and
Fixed p2 and y.
Own-Price Changes
x2
x1
x2 0*? x y
p1 1
*?
p1 = p1’ < p2
’
Fixed p2 and y.
Own-Price Changes
x2
x1
p1
x1*
x2 0*? x y
p1 1
*?
p1’
p1 = p1’ < p2
’
x yp1
1
*?
’
Fixed p2 and y.
Own-Price Changes
x2
x1
p1
x1*
p1’
p1 = p1’’ = p2
Fixed p2 and y.
Own-Price Changes
x2
x1
p1
x1*
p1’
p1 = p1’’ = p2
Fixed p2 and y.
Own-Price Changes
x2
x1
p1
x1*
x2 0*? x y
p1 1
*?
p1’
p1 = p1’’ = p2
’’
x1 0*?
x yp2
2
*?
Fixed p2 and y.
Own-Price Changes
x2
x1
p1
x1*
x2 0*? x y
p1 2
*?
p1’
p1 = p1’’ = p2
x1 0*?
x yp2
2
*?
0 1
2
x yp*
p2 = p1’’
Fixed p2 and y.
Own-Price Changes
x2
x1
p1
x1*
x yp2
2
*?
x1 0*?
p1’
p1’’’
x1 0*?
p2 = p1’’
Fixed p2 and y.
Own-Price Changes
x2
x1
p1
x1*
p1’
p2 = p1’’
p1’’’ x y
p1 1
*?
0 1
2
x yp*
y
p2
p1 price
offer
curve
Ordinary
demand curve
for commodity 1
Own-Price Changes
Usually we ask,Given the price for
commodity 1 what is the quantity
demanded of commodity 1?”
But we could also ask the inverse
question,At what price for
commodity 1 would a given quantity
of commodity 1 be demanded?”
Own-Price Changes
p1
x1*
p1’
Given p1’,what quantity is
demanded of commodity 1?
Own-Price Changes
p1
x1*
p1’
Given p1’,what quantity is
demanded of commodity 1?
Answer,x1’ units.
x1’
Own-Price Changes
p1
x1*x1’
Given p1’,what quantity is
demanded of commodity 1?
Answer,x1’ units.
The inverse question is:
Given x1’ units are
demanded,what is the
price of
commodity 1?
Own-Price Changes
p1
x1*
p1’
x1’
Given p1’,what quantity is
demanded of commodity 1?
Answer,x1’ units.
The inverse question is:
Given x1’ units are
demanded,what is the
price of
commodity 1?
Answer,p1’
Own-Price Changes
Taking quantity demanded as given
and then asking what must be price
describes the inverse demand
function of a commodity.
Own-Price Changes
A Cobb-Douglas example:
x ay
a b p1 1
*
( )
is the ordinary demand function andp ay
a b x
1
1
( ) *
is the inverse demand function.
Own-Price Changes
A perfect-complements example:
x y
p p1 1 2
*?
is the ordinary demand function andp y
x
p1
1
2*
is the inverse demand function.
Income Changes
How does the value of x1*(p1,p2,y)
change as y changes,holding both
p1 and p2 constant?
x
2
x
1
Income Changes
Fixed p1 and p2.
y’ < y’’ < y’’’
x
2
x
1
Income Changes
Fixed p1 and p2.
y’ < y’’ < y’’’
x
2
x
1
Income Changes
Fixed p1 and p2.
y’ < y’’ < y’’’
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
x
2
x
1
Income Changes
Fixed p1 and p2.
y’ < y’’ < y’’’
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
Income
offer curve
Income Changes
A plot of quantity demanded against
income is called an Engel curve.
x
2
x
1
Income Changes
Fixed p1 and p2.
y’ < y’’ < y’’’
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
Income
offer curve
x
2
x
1
Income Changes
Fixed p1 and p2.
y’ < y’’ < y’’’
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
Income
offer curve
x1*
y
x1’’’
x1’’
x1’
y’
y’’
y’’’
x
2
x
1
Income Changes
Fixed p1 and p2.
y’ < y’’ < y’’’
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
Income
offer curve
x1*
y
x1’’’
x1’’
x1’
y’
y’’
y’’’ Engel
curve;
good 1
x
2
x
1
Income Changes
Fixed p1 and p2.
y’ < y’’ < y’’’
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
Income
offer curve x2*
y
x2’’’
x2’’
x2’
y’
y’’
y’’’
x
2
x
1
Income Changes
Fixed p1 and p2.
y’ < y’’ < y’’’
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
Income
offer curve x2*
y
x2’’’
x2’’
x2’
y’
y’’
y’’’
Engel
curve;
good 2
x
2
x
1
Income Changes
Fixed p1 and p2.
y’ < y’’ < y’’’
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
Income
offer curve
x1*
x2*
y
y
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
y’
y’’
y’’’
y’
y’’
y’’’
Engel
curve;
good 2
Engel
curve;
good 1
Income Changes and Cobb-
Douglas Preferences
An example of computing the
equations of Engel curves; the Cobb-
Douglas case.
The ordinary demand equations are
U x x x xa b(,),1 2 1 2?
x ay
a b p
x by
a b p1 1 2 2
* *
( );
( )
.?
Income Changes and Cobb-
Douglas Preferences
x ay
a b p
x by
a b p1 1 2 2
* *
( );
( )
.?
Rearranged to isolate y,these are:
y
a b p
a
x
y
a b p
b
x
( )
( )
*
*
1
1
2
2
Engel curve for good 1
Engel curve for good 2
Income Changes and Cobb-
Douglas Preferences
y
y x1*
x2*
y a b pa x( ) *1 1
Engel curve
for good 1
y a b pb x( ) *2 2
Engel curve
for good 2
Income Changes and Perfectly-
Complementary Preferences
Another example of computing the
equations of Engel curves; the
perfectly-complementary case.
The ordinary demand equations are
x x y
p p1 2 1 2
* *,
U x x x x(,) m i n,.1 2 1 2?
Income Changes and Perfectly-
Complementary Preferences
Rearranged to isolate y,these are:y p p x
y p p x
( )
( )
*
*
1 2 1
1 2 2
Engel curve for good 1
x x y
p p1 2 1 2
* *,
Engel curve for good 2
Fixed p1 and p2.
Income Changes
x1
x2
Income Changes
x1
x2
y’ < y’’ < y’’’
Fixed p1 and p2.
Income Changes
x1
x2
y’ < y’’ < y’’’
Fixed p1 and p2.
Income Changes
x1
x2
y’ < y’’ < y’’’
x1’’
x1’
x2’’’
x2’’
x2’
x1’’’
Fixed p1 and p2.
Income Changes
x1
x2
y’ < y’’ < y’’’
x1’’
x1’
x2’’’
x2’’
x2’
x1’’’ x1*
y
y’
y’’
y’’’ Engel
curve;
good 1
x1’’’
x1’’
x1’
Fixed p1 and p2.
Income Changes
x1
x2
y’ < y’’ < y’’’
x1’’
x1’
x2’’’
x2’’
x2’
x1’’’
x2*
y
x2’’’
x2’’
x2’
y’
y’’
y’’’
Engel
curve;
good 2Fixed p1 and p2.
Income Changes
x1
x2
y’ < y’’ < y’’’
x1’’
x1’
x2’’’
x2’’
x2’
x1’’’ x1*
x2*
y
y x2’’’x
2’’
x2’
y’
y’’
y’’’
y’
y’’
y’’’
Engel
curve;
good 2
Engel
curve;
good 1
x1’’’
x1’’
x1’
Fixed p1 and p2.
Income Changes
x1*
x2*
y
y x2’’’x
2’’
x2’
y’
y’’
y’’’
y’
y’’
y’’’
x1’’’
x1’’
x1’
y p p x( ) *1 2 2
y p p x( ) *1 2 1
Engel
curve;
good 2
Engel
curve;
good 1
Fixed p1 and p2.
Income Changes and Perfectly-
Substitutable Preferences
Another example of computing the
equations of Engel curves; the
perfectly-substitution case.
The ordinary demand equations are
U x x x x(,),1 2 1 2
Income Changes and Perfectly-
Substitutable Preferences
x p p y
if p p
y p if p p1 1 2
1 2
1 1 2
0*
(,,)
,
/,
x p p y
if p p
y p if p p2 1 2
1 2
2 1 2
0*
(,,)
,
/,.
Income Changes and Perfectly-
Substitutable Preferences
x p p y
if p p
y p if p p1 1 2
1 2
1 1 2
0*
(,,)
,
/,
x p p y
if p p
y p if p p2 1 2
1 2
2 1 2
0*
(,,)
,
/,.
Suppose p1 < p2,Then
Income Changes and Perfectly-
Substitutable Preferences
x p p y
if p p
y p if p p1 1 2
1 2
1 1 2
0*
(,,)
,
/,
x p p y
if p p
y p if p p2 1 2
1 2
2 1 2
0*
(,,)
,
/,.
Suppose p1 < p2,Then x
y
p1 1
*?
x 2 0*?and
Income Changes and Perfectly-
Substitutable Preferences
x p p y
if p p
y p if p p1 1 2
1 2
1 1 2
0*
(,,)
,
/,
x p p y
if p p
y p if p p2 1 2
1 2
2 1 2
0*
(,,)
,
/,.
Suppose p1 < p2,Then x
y
p1 1
*?
x 2 0*?and
x 2 0*,?y p x? 1 1and
Income Changes and Perfectly-
Substitutable Preferences
x 2 0*,?y p x? 1 1*
y y
x1* x2*0
Engel curve
for good 1
Engel curve
for good 2
Income Changes
In every example so far the Engel
curves have all been straight lines?
Q,Is this true in general?
A,No,Engel curves are straight
lines if the consumer’s preferences
are homothetic.
Homotheticity （ 位似偏好）
A consumer’s preferences are
homothetic if and only if
for every k > 0.
That is,the consumer’s MRS is the
same anywhere on a straight line
drawn from the origin.
(x1,x2) (y1,y2) (kx1,kx2) (ky1,ky2)pp
Income Effects -- A
Nonhomothetic Example
Quasilinear preferences are not
homothetic.
For example,
Optimal interior consumption:
1 2 1 2(,) ( ),U x x v x x
U x x x x(,),1 2 1 2
1
1
2
'( * ),
p
vx
p
Quasi-linear Indifference Curves
x2
x1
Each curve is a vertically shifted
copy of the others.
Each curve intersects
both axes.
Income Changes; Quasilinear
Utility
x2
x1
x1~
Income Changes; Quasilinear
Utility
x2
x1
x1~
x1*
y
x1~
Engel
curve
for
good 1
Income Changes; Quasilinear
Utility
x2
x1
x1~
x2*
y Engel
curve
for
good 2
Income Changes; Quasilinear
Utility
x2
x1
x1~
x1*
x2*
y
y
x1~
Engel
curve
for
good 2
Engel
curve
for
good 1
Income Effects
A good for which quantity demanded
rises with income is called normal （
正常品）,
Therefore a normal good’s Engel
curve is positively sloped.
Income Effects
A good for which quantity demanded
falls as income increases is called
income inferior （ 劣质品）,
Therefore an income inferior good’s
Engel curve is negatively sloped.
x
2
x
1
Income Changes; Goods
1 & 2 Normal
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
Income
offer curve
x1*
x2*
y
y
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
y’
y’’
y’’’
y’
y’’
y’’’
Engel
curve;
good 2
Engel
curve;
good 1
Income Changes; Good 2 Is Normal,
Good 1 Becomes Income Inferior
x2
x1
Income Changes; Good 2 Is Normal,
Good 1 Becomes Income Inferior
x2
x1
Income Changes; Good 2 Is Normal,
Good 1 Becomes Income Inferior
x2
x1
Income Changes; Good 2 Is Normal,
Good 1 Becomes Income Inferior
x2
x1
Income Changes; Good 2 Is Normal,
Good 1 Becomes Income Inferior
x2
x1
Income Changes; Good 2 Is Normal,
Good 1 Becomes Income Inferior
x2
x1
Income
offer curve
Income Changes; Good 2 Is Normal,
Good 1 Becomes Income Inferior
x2
x1 x1*
y
Engel curve
for good 1
Income Changes; Good 2 Is Normal,
Good 1 Becomes Income Inferior
x2
x1 x1*
x2*
y
y
Engel curve
for good 2
Engel curve
for good 1
Ordinary Goods （ 一般商品）
A good is called ordinary if the
quantity demanded of it always
increases as its own price decreases.
Ordinary Goods
Fixed p2 and y.
x1
x2
Ordinary Goods
Fixed p2 and y.
x1
x2
p1 price
offer
curve
Ordinary Goods
Fixed p2 and y.
x1
x2
p1 price
offer
curve
x1*
Downward-sloping
demand curve
Good 1 is
ordinary
p1
Giffen Goods （ 吉芬商品）
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.
Ordinary Goods
Fixed p2 and y.
x1
x2
Ordinary Goods
Fixed p2 and y.
x1
x2 p
1 price offer
curve
Ordinary Goods
Fixed p2 and y.
x1
x2 p
1 price offer
curve
x1*
Demand curve has
a positively
sloped part
Good 1 is
Giffen
p1
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.
Cross-Price Effects
A perfect-complements example:
x y
p p1 1 2
*?
x
p
y
p p
1
2 1 2 2
0
*
.
so
Therefore commodity 2 is a gross
complement for commodity 1.
Cross-Price Effects
p1
x1*
p1’
p1’’
p1’’’
y
p2’
Increase the price of
good 2 from p2’ to p2’’
and
Cross-Price Effects
p1
x1*
p1’
p1’’
p1’’’
y
p2’’
Increase the price of
good 2 from p2’ to p2’’
and the demand curve
for good 1 shifts inwards
-- good 2 is a
complement for good 1,
Cross-Price Effects
A Cobb- Douglas example:
x by
a b p2 2
*
( )
so
Cross-Price Effects
A Cobb- Douglas example:
x by
a b p2 2
*
( )
x
p
2
1
0
*
.?so
Therefore commodity 1 is neither a gross
complement nor a gross substitute for
commodity 2.
Demand
Properties of Demand Functions
Comparative statics analysis （ 比较静态分析） of ordinary demand
functions -- the study of how
ordinary demands x1*(p1,p2,y) and
x2*(p1,p2,y) change as prices p1,p2
and income y change.
Structure
Own-price changes
– Price offer curve （ 价格提供曲线）
– Ordinary demand curve
– Inverse demand curve （ 反需求函数）
Income changes
– Income offer curve （ 收入提供曲线）
– Engel curve （ 恩格尔曲线）
Cross-price effects
Own-Price Changes
How does x1*(p1,p2,y) change as p1
changes,holding p2 and y constant?
Suppose only p1 increases,from p1’
to p1’’ and then to p1’’’.
x1
x2
p1 = p1’
Fixed p2 and y.
p1x1 + p2x2 = y
Own-Price Changes
Own-Price Changes
x1
x2
p1= p1’’
p1 = p1’
Fixed p2 and y.
p1x1 + p2x2 = y
Own-Price Changes
x1
x2
p1= p1’’
p1=
p1’’’
Fixed p2 and y.
p1 = p1’
p1x1 + p2x2 = y
x
2
x
1
p1 = p1’
Own-Price Changes
Fixed p2 and y.
x
2
x
1
x1*(p1’)
Own-Price Changes
p1 = p1’
Fixed p2 and y.
x
2
x
1
x1*(p1’)
p1
x1*(p1’)
p1’
x1*
Own-Price Changes
Fixed p2 and y.
p1 = p1’
x
2
x
1
x1*(p1’)
p1
x1*(p1’)
p1’
p1 = p1’’
x1*
Own-Price Changes
Fixed p2 and y.
x
2
x
1
x1*(p1’)
x1*(p1’’)
p1
x1*(p1’)
p1’
p1 = p1’’
x1*
Own-Price Changes
Fixed p2 and y.
x
2
x
1
x1*(p1’)
x1*(p1’’)
p1
x1*(p1’)
x1*(p1’’)
p1’
p1’’
x1*
Own-Price Changes
Fixed p2 and y.
x
2
x
1
x1*(p1’)
x1*(p1’’)
p1
x1*(p1’)
x1*(p1’’)
p1’
p1’’
p1 = p1’’’
x1*
Own-Price Changes
Fixed p2 and y.
x
2
x
1
x1*(p1’’’) x1*(p1’)
x1*(p1’’)
p1
x1*(p1’)
x1*(p1’’)
p1’
p1’’
p1 = p1’’’
x1*
Own-Price Changes
Fixed p2 and y.
x
2
x
1
x1*(p1’’’) x1*(p1’)
x1*(p1’’)
p1
x1*(p1’)x1*(p1’’’)
x1*(p1’’)
p1’
p1’’
p1’’’
x1*
Own-Price Changes
Fixed p2 and y.
x
2
x
1
x1*(p1’’’) x1*(p1’)
x1*(p1’’)
p1
x1*(p1’)x1*(p1’’’)
x1*(p1’’)
p1’
p1’’
p1’’’
x1*
Own-Price Changes Ordinarydemand curve
for commodity 1Fixed p2 and y.
x
2
x
1
x1*(p1’’’) x1*(p1’)
x1*(p1’’)
p1
x1*(p1’)x1*(p1’’’)
x1*(p1’’)
p1’
p1’’
p1’’’
x1*
Own-Price Changes Ordinarydemand curve
for commodity 1Fixed p2 and y.
x
2
x
1
x1*(p1’’’) x1*(p1’)
x1*(p1’’)
p1
x1*(p1’)x1*(p1’’’)
x1*(p1’’)
p1’
p1’’
p1’’’
x1*
Own-Price Changes Ordinarydemand curve
for commodity 1
p1 price
offer
curve
Fixed p2 and y.
Own-Price Changes
The curve containing all the utility-
maximizing 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 Cobb-Douglas preferences?
Own-Price Changes
What does a p1 price-offer curve look
like for Cobb-Douglas preferences?
Take
Then the ordinary demand functions
for commodities 1 and 2 are
U x x x xa b(,),1 2 1 2?
Own-Price Changes
x p p y a
a b
y
p1 1 2 1
* (,,)?
x p p y b
a b
y
p2 1 2 2
* (,,),?
and
Notice that x2* does not vary with p1 so the
p1 price offer curve is
Own-Price Changes
x p p y a
a b
y
p1 1 2 1
* (,,)?
x p p y b
a b
y
p2 1 2 2
* (,,),?
and
Notice that x2* does not vary with p1 so the
p1 price offer curve is flat
Own-Price Changes
x p p y a
a b
y
p1 1 2 1
* (,,)?
x p p y b
a b
y
p2 1 2 2
* (,,),?
and
Notice that x2* does not vary with p1 so the
p1 price offer curve is flat and the ordinary
demand curve for commodity 1 is a
Own-Price Changes
x p p y a
a b
y
p1 1 2 1
* (,,)?
x p p y b
a b
y
p2 1 2 2
* (,,),?
and
Notice that x2* does not vary with p1 so the
p1 price offer curve is flat and the ordinary
demand curve for commodity 1 is a
rectangular hyperbola.
x1*(p1’’’) x1*(p1’)
x1*(p1’’)
x
2
x
1
Own-Price Changes
Fixed p2 and y.x
by
a b p
2
2
*
( )
x aya b p1
1
*
( )
x1*(p1’’’) x1*(p1’)
x1*(p1’’)
x
2
x
1
p1
x1*
Own-Price Changes Ordinarydemand curve
for commodity 1
is
Fixed p2 and y.x
by
a b p
2
2
*
( )
x aya b p1
1
*
( )
x aya b p1
1
*
( )
Own-Price Changes
What does a p1 price-offer curve look
like for a perfect-complements utility
function?
Own-Price Changes
What does a p1 price-offer curve look
like for a perfect-complements utility
function?
U x x x x(,) m i n,.1 2 1 2?
Then the ordinary demand functions
for commodities 1 and 2 are
Own-Price Changes
x p p y x p p y y
p p1 1 2 2 1 2 1 2
* *(,,) (,,),
Own-Price Changes
x p p y x p p y y
p p1 1 2 2 1 2 1 2
* *(,,) (,,),
With p2 and y fixed,higher p1 causes
smaller x1* and x2*.
Own-Price Changes
x p p y x p p y y
p p1 1 2 2 1 2 1 2
* *(,,) (,,),
With p2 and y fixed,higher p1 causes
smaller x1* and x2*.
p x x y
p1 1 2 2
0,.* *As
Own-Price Changes
x p p y x p p y y
p p1 1 2 2 1 2 1 2
* *(,,) (,,),
With p2 and y fixed,higher p1 causes
smaller x1* and x2*.
p x x y
p1 1 2 2
0,.* *As
p x x1 1 2 0,.* *As
Fixed p2 and y.
Own-Price Changes
x1
x2
p1
x1*
Fixed p2 and y.x
y
p p
2
1 2
*?
x yp p1
1 2
*?
Own-Price Changes
x1
x2
p1’
x yp p1
1 2
*?
’
p1 = p1’
’
’
y/p2
p1
x1*
Fixed p2 and y.x
y
p p
2
1 2
*?
x yp p1
1 2
*?
Own-Price Changes
x1
x2
p1’
p1’’
p1 = p1’’
’’
x yp p1
1 2
*?
’’
’’
y/p2
p1
x1*
Fixed p2 and y.
x
y
p p
2
1 2
*?
x yp p1
1 2
*?
Own-Price Changes
x1
x2
p1’
p1’’
p1’’’
x yp p1
1 2
*?
p1 = p1’’’
’’’
’’’
’’’
y/p2
p1
x1*
Ordinary
demand curve
for commodity 1
is
Fixed p2 and y.x
y
p p
2
1 2
*?
x yp p1
1 2
*?
x yp p1
1 2
*,?
Own-Price Changes
x1
x2
p1’
p1’’
p1’’’
y
p2
y/p2
Own-Price Changes
What does a p1 price-offer curve look
like for a perfect-substitutes utility
function?
U x x x x(,),1 2 1 2
Then the ordinary demand functions
for commodities 1 and 2 are
Own-Price Changes
x p p y
if p p
y p if p p1 1 2
1 2
1 1 2
0*
(,,)
,
/,
x p p y
if p p
y p if p p2 1 2
1 2
2 1 2
0*
(,,)
,
/,.
and
Fixed p2 and y.
Own-Price Changes
x2
x1
x2 0*? x y
p1 1
*?
p1 = p1’ < p2
’
Fixed p2 and y.
Own-Price Changes
x2
x1
p1
x1*
x2 0*? x y
p1 1
*?
p1’
p1 = p1’ < p2
’
x yp1
1
*?
’
Fixed p2 and y.
Own-Price Changes
x2
x1
p1
x1*
p1’
p1 = p1’’ = p2
Fixed p2 and y.
Own-Price Changes
x2
x1
p1
x1*
p1’
p1 = p1’’ = p2
Fixed p2 and y.
Own-Price Changes
x2
x1
p1
x1*
x2 0*? x y
p1 1
*?
p1’
p1 = p1’’ = p2
’’
x1 0*?
x yp2
2
*?
Fixed p2 and y.
Own-Price Changes
x2
x1
p1
x1*
x2 0*? x y
p1 2
*?
p1’
p1 = p1’’ = p2
x1 0*?
x yp2
2
*?
0 1
2
x yp*
p2 = p1’’
Fixed p2 and y.
Own-Price Changes
x2
x1
p1
x1*
x yp2
2
*?
x1 0*?
p1’
p1’’’
x1 0*?
p2 = p1’’
Fixed p2 and y.
Own-Price Changes
x2
x1
p1
x1*
p1’
p2 = p1’’
p1’’’ x y
p1 1
*?
0 1
2
x yp*
y
p2
p1 price
offer
curve
Ordinary
demand curve
for commodity 1
Own-Price Changes
Usually we ask,Given the price for
commodity 1 what is the quantity
demanded of commodity 1?”
But we could also ask the inverse
question,At what price for
commodity 1 would a given quantity
of commodity 1 be demanded?”
Own-Price Changes
p1
x1*
p1’
Given p1’,what quantity is
demanded of commodity 1?
Own-Price Changes
p1
x1*
p1’
Given p1’,what quantity is
demanded of commodity 1?
Answer,x1’ units.
x1’
Own-Price Changes
p1
x1*x1’
Given p1’,what quantity is
demanded of commodity 1?
Answer,x1’ units.
The inverse question is:
Given x1’ units are
demanded,what is the
price of
commodity 1?
Own-Price Changes
p1
x1*
p1’
x1’
Given p1’,what quantity is
demanded of commodity 1?
Answer,x1’ units.
The inverse question is:
Given x1’ units are
demanded,what is the
price of
commodity 1?
Answer,p1’
Own-Price Changes
Taking quantity demanded as given
and then asking what must be price
describes the inverse demand
function of a commodity.
Own-Price Changes
A Cobb-Douglas example:
x ay
a b p1 1
*
( )
is the ordinary demand function andp ay
a b x
1
1
( ) *
is the inverse demand function.
Own-Price Changes
A perfect-complements example:
x y
p p1 1 2
*?
is the ordinary demand function andp y
x
p1
1
2*
is the inverse demand function.
Income Changes
How does the value of x1*(p1,p2,y)
change as y changes,holding both
p1 and p2 constant?
x
2
x
1
Income Changes
Fixed p1 and p2.
y’ < y’’ < y’’’
x
2
x
1
Income Changes
Fixed p1 and p2.
y’ < y’’ < y’’’
x
2
x
1
Income Changes
Fixed p1 and p2.
y’ < y’’ < y’’’
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
x
2
x
1
Income Changes
Fixed p1 and p2.
y’ < y’’ < y’’’
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
Income
offer curve
Income Changes
A plot of quantity demanded against
income is called an Engel curve.
x
2
x
1
Income Changes
Fixed p1 and p2.
y’ < y’’ < y’’’
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
Income
offer curve
x
2
x
1
Income Changes
Fixed p1 and p2.
y’ < y’’ < y’’’
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
Income
offer curve
x1*
y
x1’’’
x1’’
x1’
y’
y’’
y’’’
x
2
x
1
Income Changes
Fixed p1 and p2.
y’ < y’’ < y’’’
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
Income
offer curve
x1*
y
x1’’’
x1’’
x1’
y’
y’’
y’’’ Engel
curve;
good 1
x
2
x
1
Income Changes
Fixed p1 and p2.
y’ < y’’ < y’’’
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
Income
offer curve x2*
y
x2’’’
x2’’
x2’
y’
y’’
y’’’
x
2
x
1
Income Changes
Fixed p1 and p2.
y’ < y’’ < y’’’
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
Income
offer curve x2*
y
x2’’’
x2’’
x2’
y’
y’’
y’’’
Engel
curve;
good 2
x
2
x
1
Income Changes
Fixed p1 and p2.
y’ < y’’ < y’’’
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
Income
offer curve
x1*
x2*
y
y
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
y’
y’’
y’’’
y’
y’’
y’’’
Engel
curve;
good 2
Engel
curve;
good 1
Income Changes and Cobb-
Douglas Preferences
An example of computing the
equations of Engel curves; the Cobb-
Douglas case.
The ordinary demand equations are
U x x x xa b(,),1 2 1 2?
x ay
a b p
x by
a b p1 1 2 2
* *
( );
( )
.?
Income Changes and Cobb-
Douglas Preferences
x ay
a b p
x by
a b p1 1 2 2
* *
( );
( )
.?
Rearranged to isolate y,these are:
y
a b p
a
x
y
a b p
b
x
( )
( )
*
*
1
1
2
2
Engel curve for good 1
Engel curve for good 2
Income Changes and Cobb-
Douglas Preferences
y
y x1*
x2*
y a b pa x( ) *1 1
Engel curve
for good 1
y a b pb x( ) *2 2
Engel curve
for good 2
Income Changes and Perfectly-
Complementary Preferences
Another example of computing the
equations of Engel curves; the
perfectly-complementary case.
The ordinary demand equations are
x x y
p p1 2 1 2
* *,
U x x x x(,) m i n,.1 2 1 2?
Income Changes and Perfectly-
Complementary Preferences
Rearranged to isolate y,these are:y p p x
y p p x
( )
( )
*
*
1 2 1
1 2 2
Engel curve for good 1
x x y
p p1 2 1 2
* *,
Engel curve for good 2
Fixed p1 and p2.
Income Changes
x1
x2
Income Changes
x1
x2
y’ < y’’ < y’’’
Fixed p1 and p2.
Income Changes
x1
x2
y’ < y’’ < y’’’
Fixed p1 and p2.
Income Changes
x1
x2
y’ < y’’ < y’’’
x1’’
x1’
x2’’’
x2’’
x2’
x1’’’
Fixed p1 and p2.
Income Changes
x1
x2
y’ < y’’ < y’’’
x1’’
x1’
x2’’’
x2’’
x2’
x1’’’ x1*
y
y’
y’’
y’’’ Engel
curve;
good 1
x1’’’
x1’’
x1’
Fixed p1 and p2.
Income Changes
x1
x2
y’ < y’’ < y’’’
x1’’
x1’
x2’’’
x2’’
x2’
x1’’’
x2*
y
x2’’’
x2’’
x2’
y’
y’’
y’’’
Engel
curve;
good 2Fixed p1 and p2.
Income Changes
x1
x2
y’ < y’’ < y’’’
x1’’
x1’
x2’’’
x2’’
x2’
x1’’’ x1*
x2*
y
y x2’’’x
2’’
x2’
y’
y’’
y’’’
y’
y’’
y’’’
Engel
curve;
good 2
Engel
curve;
good 1
x1’’’
x1’’
x1’
Fixed p1 and p2.
Income Changes
x1*
x2*
y
y x2’’’x
2’’
x2’
y’
y’’
y’’’
y’
y’’
y’’’
x1’’’
x1’’
x1’
y p p x( ) *1 2 2
y p p x( ) *1 2 1
Engel
curve;
good 2
Engel
curve;
good 1
Fixed p1 and p2.
Income Changes and Perfectly-
Substitutable Preferences
Another example of computing the
equations of Engel curves; the
perfectly-substitution case.
The ordinary demand equations are
U x x x x(,),1 2 1 2
Income Changes and Perfectly-
Substitutable Preferences
x p p y
if p p
y p if p p1 1 2
1 2
1 1 2
0*
(,,)
,
/,
x p p y
if p p
y p if p p2 1 2
1 2
2 1 2
0*
(,,)
,
/,.
Income Changes and Perfectly-
Substitutable Preferences
x p p y
if p p
y p if p p1 1 2
1 2
1 1 2
0*
(,,)
,
/,
x p p y
if p p
y p if p p2 1 2
1 2
2 1 2
0*
(,,)
,
/,.
Suppose p1 < p2,Then
Income Changes and Perfectly-
Substitutable Preferences
x p p y
if p p
y p if p p1 1 2
1 2
1 1 2
0*
(,,)
,
/,
x p p y
if p p
y p if p p2 1 2
1 2
2 1 2
0*
(,,)
,
/,.
Suppose p1 < p2,Then x
y
p1 1
*?
x 2 0*?and
Income Changes and Perfectly-
Substitutable Preferences
x p p y
if p p
y p if p p1 1 2
1 2
1 1 2
0*
(,,)
,
/,
x p p y
if p p
y p if p p2 1 2
1 2
2 1 2
0*
(,,)
,
/,.
Suppose p1 < p2,Then x
y
p1 1
*?
x 2 0*?and
x 2 0*,?y p x? 1 1and
Income Changes and Perfectly-
Substitutable Preferences
x 2 0*,?y p x? 1 1*
y y
x1* x2*0
Engel curve
for good 1
Engel curve
for good 2
Income Changes
In every example so far the Engel
curves have all been straight lines?
Q,Is this true in general?
A,No,Engel curves are straight
lines if the consumer’s preferences
are homothetic.
Homotheticity （ 位似偏好）
A consumer’s preferences are
homothetic if and only if
for every k > 0.
That is,the consumer’s MRS is the
same anywhere on a straight line
drawn from the origin.
(x1,x2) (y1,y2) (kx1,kx2) (ky1,ky2)pp
Income Effects -- A
Nonhomothetic Example
Quasilinear preferences are not
homothetic.
For example,
Optimal interior consumption:
1 2 1 2(,) ( ),U x x v x x
U x x x x(,),1 2 1 2
1
1
2
'( * ),
p
vx
p
Quasi-linear Indifference Curves
x2
x1
Each curve is a vertically shifted
copy of the others.
Each curve intersects
both axes.
Income Changes; Quasilinear
Utility
x2
x1
x1~
Income Changes; Quasilinear
Utility
x2
x1
x1~
x1*
y
x1~
Engel
curve
for
good 1
Income Changes; Quasilinear
Utility
x2
x1
x1~
x2*
y Engel
curve
for
good 2
Income Changes; Quasilinear
Utility
x2
x1
x1~
x1*
x2*
y
y
x1~
Engel
curve
for
good 2
Engel
curve
for
good 1
Income Effects
A good for which quantity demanded
rises with income is called normal （
正常品）,
Therefore a normal good’s Engel
curve is positively sloped.
Income Effects
A good for which quantity demanded
falls as income increases is called
income inferior （ 劣质品）,
Therefore an income inferior good’s
Engel curve is negatively sloped.
x
2
x
1
Income Changes; Goods
1 & 2 Normal
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
Income
offer curve
x1*
x2*
y
y
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
y’
y’’
y’’’
y’
y’’
y’’’
Engel
curve;
good 2
Engel
curve;
good 1
Income Changes; Good 2 Is Normal,
Good 1 Becomes Income Inferior
x2
x1
Income Changes; Good 2 Is Normal,
Good 1 Becomes Income Inferior
x2
x1
Income Changes; Good 2 Is Normal,
Good 1 Becomes Income Inferior
x2
x1
Income Changes; Good 2 Is Normal,
Good 1 Becomes Income Inferior
x2
x1
Income Changes; Good 2 Is Normal,
Good 1 Becomes Income Inferior
x2
x1
Income Changes; Good 2 Is Normal,
Good 1 Becomes Income Inferior
x2
x1
Income
offer curve
Income Changes; Good 2 Is Normal,
Good 1 Becomes Income Inferior
x2
x1 x1*
y
Engel curve
for good 1
Income Changes; Good 2 Is Normal,
Good 1 Becomes Income Inferior
x2
x1 x1*
x2*
y
y
Engel curve
for good 2
Engel curve
for good 1
Ordinary Goods （ 一般商品）
A good is called ordinary if the
quantity demanded of it always
increases as its own price decreases.
Ordinary Goods
Fixed p2 and y.
x1
x2
Ordinary Goods
Fixed p2 and y.
x1
x2
p1 price
offer
curve
Ordinary Goods
Fixed p2 and y.
x1
x2
p1 price
offer
curve
x1*
Downward-sloping
demand curve
Good 1 is
ordinary
p1
Giffen Goods （ 吉芬商品）
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.
Ordinary Goods
Fixed p2 and y.
x1
x2
Ordinary Goods
Fixed p2 and y.
x1
x2 p
1 price offer
curve
Ordinary Goods
Fixed p2 and y.
x1
x2 p
1 price offer
curve
x1*
Demand curve has
a positively
sloped part
Good 1 is
Giffen
p1
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.
Cross-Price Effects
A perfect-complements example:
x y
p p1 1 2
*?
x
p
y
p p
1
2 1 2 2
0
*
.
so
Therefore commodity 2 is a gross
complement for commodity 1.
Cross-Price Effects
p1
x1*
p1’
p1’’
p1’’’
y
p2’
Increase the price of
good 2 from p2’ to p2’’
and
Cross-Price Effects
p1
x1*
p1’
p1’’
p1’’’
y
p2’’
Increase the price of
good 2 from p2’ to p2’’
and the demand curve
for good 1 shifts inwards
-- good 2 is a
complement for good 1,
Cross-Price Effects
A Cobb- Douglas example:
x by
a b p2 2
*
( )
so
Cross-Price Effects
A Cobb- Douglas example:
x by
a b p2 2
*
( )
x
p
2
1
0
*
.?so
Therefore commodity 1 is neither a gross
complement nor a gross substitute for
commodity 2.
