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6
Demand

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Properties of Demand Functions
uComparative statics analysis 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.

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Own-Price Changes
uHow does x1*(p1,p2,y) change as p1 changes, holding p2 and y constant?
uSuppose only p1 increases, from p1’ to p1’’ and then to p1’’’.

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x1
x2
p1 = p1’
Fixed p2 and y.
p1x1 + p2x2 = y
Own-Price Changes

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Own-Price Changes
x1
x2
p1= p1’’
p1 = p1’
Fixed p2 and y.
p1x1 + p2x2 = y

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Own-Price Changes
x1
x2
p1= p1’’
p1=
p1’’’
Fixed p2 and y.
p1 = p1’
p1x1 + p2x2 = y

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Own-Price Changes
Fixed p2 and y.
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x1*(p1’)
Own-Price Changes
p1 = p1’
Fixed p2 and y.
06-07.png

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x1*(p1’)
p1
x1*(p1’)
p1’
x1*
Own-Price Changes
Fixed p2 and y.
p1 = p1’
06-07.png

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x1*(p1’)
p1
x1*(p1’)
p1’
p1 = p1’’
x1*
Own-Price Changes
Fixed p2 and y.
06-07.png

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‹#›
x1*(p1’)
x1*(p1’’)
p1
x1*(p1’)
p1’
p1 = p1’’
x1*
Own-Price Changes
Fixed p2 and y.
06-07.png

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‹#›
x1*(p1’)
x1*(p1’’)
p1
x1*(p1’)
x1*(p1’’)
p1’
p1’’
x1*
Own-Price Changes
Fixed p2 and y.
06-07.png

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‹#›
x1*(p1’)
x1*(p1’’)
p1
x1*(p1’)
x1*(p1’’)
p1’
p1’’
p1 = p1’’’
x1*
Own-Price Changes
Fixed p2 and y.
06-07.png

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‹#›
x1*(p1’’’)
x1*(p1’)
x1*(p1’’)
p1
x1*(p1’)
x1*(p1’’)
p1’
p1’’
p1 = p1’’’
x1*
Own-Price Changes
Fixed p2 and y.
06-07.png

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‹#›
x1*(p1’’’)
x1*(p1’)
x1*(p1’’)
p1
x1*(p1’)
x1*(p1’’’)
x1*(p1’’)
p1’
p1’’
p1’’’
x1*
Own-Price Changes
Fixed p2 and y.
06-07.png

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‹#›
x1*(p1’’’)
x1*(p1’)
x1*(p1’’)
p1
x1*(p1’)
x1*(p1’’’)
x1*(p1’’)
p1’
p1’’
p1’’’
x1*
Own-Price Changes
Ordinary
demand curve
for commodity 1
Fixed p2 and y.
06-07.png

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x1*(p1’’’)
x1*(p1’)
x1*(p1’’)
p1
x1*(p1’)
x1*(p1’’’)
x1*(p1’’)
p1’
p1’’
p1’’’
x1*
Own-Price Changes
Ordinary
demand curve
for commodity 1
Fixed p2 and y.
06-07.png

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‹#›
x1*(p1’’’)
x1*(p1’)
x1*(p1’’)
p1
x1*(p1’)
x1*(p1’’’)
x1*(p1’’)
p1’
p1’’
p1’’’
x1*
Own-Price Changes
Ordinary
demand curve
for commodity 1
p1 price
  offer
     curve
Fixed p2 and y.
06-07.png

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Own-Price Changes
uThe curve containing all the utility-maximizing bundles traced out as p1 changes, with p2 and y
constant, is the p1- price offer curve.
uThe plot of the x1-coordinate of the p1- price offer curve against p1 is the ordinary demand curve
for commodity 1.

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Own-Price Changes
uWhat does a p1 price-offer curve look like for Cobb-Douglas preferences?

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Own-Price Changes
uWhat does a p1 price-offer curve look like for Cobb-Douglas preferences?
uTake
Then the ordinary demand functions for commodities 1 and 2 are

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Own-Price Changes
and
Notice that x2* does not vary with p1 so the
p1 price offer curve is

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Own-Price Changes
and
Notice that x2* does not vary with p1 so the
p1 price offer curve is  flat

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Own-Price Changes
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

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Own-Price Changes
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.

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x1*(p1’’’)
x1*(p1’)
x1*(p1’’)
Own-Price Changes
Fixed p2 and y.

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‹#›
x1*(p1’’’)
x1*(p1’)
x1*(p1’’)
p1
x1*
Own-Price Changes
Ordinary
demand curve
for commodity 1
   is
Fixed p2 and y.

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‹#›
Own-Price Changes
uWhat does a p1 price-offer curve look like for a perfect-complements utility function?

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Own-Price Changes
uWhat does a p1 price-offer curve look like for a perfect-complements utility function?
Then the ordinary demand functions
for commodities 1 and 2 are

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Own-Price Changes

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Own-Price Changes
With p2 and y fixed, higher p1 causes
smaller x1* and x2*.

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Own-Price Changes
With p2 and y fixed, higher p1 causes
smaller x1* and x2*.
As

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Own-Price Changes
With p2 and y fixed, higher p1 causes
smaller x1* and x2*.
As
As

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Fixed p2 and y.
Own-Price Changes
x1
x2

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p1
x1*
Fixed p2 and y.
Own-Price Changes
x1
x2
p1’
 ’
p1 = p1’
 ’
 ’
y/p2

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p1
x1*
Fixed p2 and y.
Own-Price Changes
x1
x2
p1’
p1’’
p1 = p1’’
’’
’’
’’
y/p2

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p1
x1*
Fixed p2 and y.
Own-Price Changes
x1
x2
p1’
p1’’
p1’’’
p1 = p1’’’
’’’
’’’
’’’
y/p2

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‹#›
p1
x1*
Ordinary
demand curve
for commodity 1
   is
Fixed p2 and y.
Own-Price Changes
x1
x2
p1’
p1’’
p1’’’
y/p2

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‹#›
Own-Price Changes
uWhat does a p1 price-offer curve look like for a perfect-substitutes utility function?
Then the ordinary demand functions
for commodities 1 and 2 are

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Own-Price Changes
and

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Fixed p2 and y.
Own-Price Changes
x2
x1
Fixed p2 and y.
p1 = p1’ < p2
’

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‹#›
Fixed p2 and y.
Own-Price Changes
x2
x1
p1
x1*
Fixed p2 and y.
p1’
p1 = p1’ < p2
’
’

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‹#›
Fixed p2 and y.
Own-Price Changes
x2
x1
p1
x1*
Fixed p2 and y.
p1’
p1 = p1’’ = p2

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Fixed p2 and y.
Own-Price Changes
x2
x1
p1
x1*
Fixed p2 and y.
p1’
p1 = p1’’ = p2

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Fixed p2 and y.
Own-Price Changes
x2
x1
p1
x1*
Fixed p2 and y.
p1’
p1 = p1’’ = p2
’’

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Fixed p2 and y.
Own-Price Changes
x2
x1
p1
x1*
Fixed p2 and y.
p1’
p1 = p1’’ = p2
p2 = p1’’

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‹#›
Fixed p2 and y.
Own-Price Changes
x2
x1
p1
x1*
Fixed p2 and y.
p1’
p1’’’
p2 = p1’’

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‹#›
Fixed p2 and y.
Own-Price Changes
x2
x1
p1
x1*
Fixed p2 and y.
p1’
p2 = p1’’
p1’’’
p1 price
    offer
       curve
Ordinary
demand curve
for commodity 1

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Own-Price Changes
uUsually we ask “Given the price for commodity 1 what is the quantity demanded of commodity 1?”
uBut we could also ask the inverse question “At what price for commodity 1 would a given quantity
of commodity 1 be demanded?”

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Own-Price Changes
p1
x1*
p1’
Given p1’, what quantity is
demanded of commodity 1?

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Own-Price Changes
p1
x1*
p1’
Given p1’, what quantity is
demanded of commodity 1?
Answer:  x1’ units.
x1’

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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?

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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’

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Own-Price Changes
uTaking quantity demanded as given and then asking what must be price describes the inverse demand
function of a commodity.

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Own-Price Changes
A Cobb-Douglas example:
is the ordinary demand function and
is the inverse demand function.

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Own-Price Changes
A perfect-complements example:
is the ordinary demand function and
is the inverse demand function.

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‹#›
Income Changes
uHow does the value of x1*(p1,p2,y) change as y changes, holding both p1 and p2 constant?

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Income Changes
Fixed p1 and p2.
y’ < y’’ < y’’’
06-07.png

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Income Changes
Fixed p1 and p2.
y’ < y’’ < y’’’
06-07.png

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‹#›
Income Changes
Fixed p1 and p2.
y’ < y’’ < y’’’
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
06-07.png

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‹#›
Income Changes
Fixed p1 and p2.
y’ < y’’ < y’’’
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
Income
offer curve
06-07.png

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Income Changes
uA plot of quantity demanded against income is called an Engel curve.

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Income Changes
Fixed p1 and p2.
y’ < y’’ < y’’’
x1’’’
x1’’
x1’
x2’’’
x2’’
x2’
Income
offer curve
06-07.png

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‹#›
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’’’
06-07.png

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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
06-07.png

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‹#›
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’’’
06-07.png

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‹#›
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
06-07.png

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‹#›
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
06-07.png

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Income Changes and Cobb-Douglas Preferences
uAn example of computing the equations of Engel curves; the Cobb-Douglas case.
u
uThe ordinary demand equations are

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Income Changes and Cobb-Douglas Preferences
Rearranged to isolate y, these are:
Engel curve for good 1
Engel curve for good 2

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Income Changes and Cobb-Douglas Preferences
y
y
x1*
x2*
Engel curve
for good 1
Engel curve
for good 2

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Income Changes and Perfectly-Complementary Preferences
uAnother example of computing the equations of Engel curves; the perfectly-complementary case.
u
uThe ordinary demand equations are

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Income Changes and Perfectly-Complementary Preferences
Rearranged to isolate y, these are:
Engel curve for good 1
Engel curve for good 2

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‹#›
Fixed p1 and p2.
Income Changes
x1
x2

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Income Changes
x1
x2
y’ < y’’ < y’’’
Fixed p1 and p2.

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‹#›
Income Changes
x1
x2
y’ < y’’ < y’’’
Fixed p1 and p2.

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‹#›
Income Changes
x1
x2
y’ < y’’ < y’’’
x1’’
x1’
x2’’’
x2’’
x2’
x1’’’
Fixed p1 and p2.

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‹#›
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.

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‹#›
Income Changes
x1
x2
y’ < y’’ < y’’’
x1’’
x1’
x2’’’
x2’’
x2’
x1’’’
x2*
y
x2’’’
x2’’
x2’
y’
y’’
y’’’
Engel
curve;
good 2
Fixed p1 and p2.

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‹#›
Income Changes
x1
x2
y’ < y’’ < y’’’
x1’’
x1’
x2’’’
x2’’
x2’
x1’’’
x1*
x2*
y
y
x2’’’
x2’’
x2’
y’
y’’
y’’’
y’
y’’
y’’’
Engel
curve;
good 2
Engel
curve;
good 1
x1’’’
x1’’
x1’
Fixed p1 and p2.

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‹#›
Income Changes
x1*
x2*
y
y
x2’’’
x2’’
x2’
y’
y’’
y’’’
y’
y’’
y’’’
x1’’’
x1’’
x1’
Engel
curve;
good 2
Engel
curve;
good 1
Fixed p1 and p2.

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‹#›
Income Changes and Perfectly-Substitutable Preferences
uAnother example of computing the equations of Engel curves; the perfectly-substitution case.
u
uThe ordinary demand equations are

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‹#›
Income Changes and Perfectly-Substitutable Preferences

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Income Changes and Perfectly-Substitutable Preferences
Suppose p1 < p2.  Then

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Income Changes and Perfectly-Substitutable Preferences
Suppose p1 < p2.  Then
and

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Income Changes and Perfectly-Substitutable Preferences
Suppose p1 < p2.  Then
and
and

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‹#›
Income Changes and Perfectly-Substitutable Preferences
y
y
x1*
x2*
0
Engel curve
for good 1
Engel curve
for good 2

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Income Changes
uIn every example so far the Engel curves have all been straight lines?
Q: Is this true in general?
uA: No.  Engel curves are straight lines if the consumer’s preferences are homothetic.

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Homotheticity
uA consumer’s preferences are homothetic if and only if
for every k > 0.
uThat is, the consumer’s MRS is the same anywhere on a straight line drawn from the origin.
Û
(x1,x2)      (y1,y2)        (kx1,kx2)       (ky1,ky2)
p
p

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‹#›
Income Effects -- A Nonhomothetic Example
uQuasilinear preferences are not homothetic.
uFor example,

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Quasi-linear Indifference Curves
x2
x1
Each curve is a vertically shifted copy of the others.
Each curve intersects
both axes.

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‹#›
Income Changes; Quasilinear Utility
x2
x1
x1
~

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‹#›
Income Changes; Quasilinear Utility
x2
x1
x1
~
x1*
y
x1
~
Engel
curve
for
good 1

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‹#›
Income Changes; Quasilinear Utility
x2
x1
x1
~
x2*
y
Engel
curve
for
good 2

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© 2010 W. W. Norton & Company, Inc.
‹#›
Income Changes; Quasilinear Utility
x2
x1
x1
~
x1*
x2*
y
y
x1
~
Engel
curve
for
good 2
Engel
curve
for
good 1

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© 2010 W. W. Norton & Company, Inc.
‹#›
Income Effects
uA good for which quantity demanded rises with income is called normal.
uTherefore a normal good’s Engel curve is positively sloped.

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© 2010 W. W. Norton & Company, Inc.
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Income Effects
uA good for which quantity demanded falls as income increases is called income inferior.
uTherefore an income inferior good’s Engel curve is negatively sloped.

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‹#›
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
06-07.png

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© 2010 W. W. Norton & Company, Inc.
‹#›
Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferior
x2
x1

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© 2010 W. W. Norton & Company, Inc.
‹#›
Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferior
x2
x1

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© 2010 W. W. Norton & Company, Inc.
‹#›
Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferior
x2
x1

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© 2010 W. W. Norton & Company, Inc.
‹#›
Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferior
x2
x1

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© 2010 W. W. Norton & Company, Inc.
‹#›
Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferior
x2
x1

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© 2010 W. W. Norton & Company, Inc.
‹#›
Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferior
x2
x1
Income
offer curve

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© 2010 W. W. Norton & Company, Inc.
‹#›
Income Changes; Good 2 Is Normal, Good 1 Becomes Income Inferior
x2
x1
x1*
y
Engel curve
for good 1

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© 2010 W. W. Norton & Company, Inc.
‹#›
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

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© 2010 W. W. Norton & Company, Inc.
‹#›
Ordinary Goods
uA good is called ordinary if the quantity demanded of it always increases as its own price
decreases.

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© 2010 W. W. Norton & Company, Inc.
‹#›
Ordinary Goods
Fixed p2 and y.
x1
x2

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© 2010 W. W. Norton & Company, Inc.
‹#›
Ordinary Goods
Fixed p2 and y.
x1
x2
p1 price
    offer
        curve

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© 2010 W. W. Norton & Company, Inc.
‹#›
Ordinary Goods
Fixed p2 and y.
x1
x2
p1 price
    offer
        curve
x1*
Downward-sloping
     demand curve
Good 1 is
ordinary
p1

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© 2010 W. W. Norton & Company, Inc.
‹#›
Giffen Goods
uIf, 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.

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© 2010 W. W. Norton & Company, Inc.
‹#›
Ordinary Goods
Fixed p2 and y.
x1
x2

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© 2010 W. W. Norton & Company, Inc.
‹#›
Ordinary Goods
Fixed p2 and y.
x1
x2
p1 price offer
  curve

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© 2010 W. W. Norton & Company, Inc.
‹#›
Ordinary Goods
Fixed p2 and y.
x1
x2
p1 price offer
  curve
x1*
Demand curve has
      a positively
          sloped part
Good 1 is
Giffen
p1

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‹#›
Cross-Price Effects
uIf 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.

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‹#›
Cross-Price Effects
A perfect-complements example:
so
Therefore commodity 2 is a gross
complement for commodity 1.

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© 2010 W. W. Norton & Company, Inc.
‹#›
Cross-Price Effects
p1
x1*
p1’
p1’’
p1’’’
’
Increase the price of
good 2 from p2’ to p2’’
and

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© 2010 W. W. Norton & Company, Inc.
‹#›
Cross-Price Effects
p1
x1*
p1’
p1’’
p1’’’
’’
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.

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Cross-Price Effects
A Cobb- Douglas example:
so

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© 2010 W. W. Norton & Company, Inc.
‹#›
Cross-Price Effects
A Cobb- Douglas example:
so
Therefore commodity 1 is neither a gross
complement nor a gross substitute for
commodity 2.