Demand

Properties of Demand Functions
Comparative 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.

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

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

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

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

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

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

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

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

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

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

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.

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.

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.

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

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

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

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

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.

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

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

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?
Then the ordinary demand functions
for commodities 1 and 2 are

Own-Price Changes

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

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

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

Fixed p2 and y.
Own-Price Changes
x1
x2

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

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

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

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

Own-Price Changes
What 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

Own-Price Changes
and

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

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

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

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

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

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

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

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

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
Inverse demand function
At optimal choice
|MRS| = p1/p2
Therefore:
p1 = p2 |MRS|
If p2 = 1, then inverse demand function simply measures MRS, i.e. how much of a good 2 consumer would want to have to compensate for a small reduction in amount of good 1.

Own-Price Changes
Inverse demand function
If good 2 is money, then MRS (and inverse demand function) measure marginal willingness to pay.

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

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

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

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

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

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.

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

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’’’

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

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’’’

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

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

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

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

Income Changes and Perfectly-Complementary Preferences
Rearranged to isolate y, these are:
Engel curve for good 1
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 2
Fixed p1 and p2.

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.

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.

Income Changes and Perfectly-Substitutable Preferences
Another example of computing the equations of Engel curves; the perfectly-substitution case.
The ordinary demand equations are

Income Changes and Perfectly-Substitutable Preferences

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

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

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

Income Changes and Perfectly-Substitutable Preferences
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
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)
p
p

Income Effects -- A Nonhomothetic Example
Quasilinear preferences are not homothetic.
For example,

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

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

Income Changes; Quasilinear Utility
x2
x1
x2*
y
Engel
curve
for
good 2

Income Changes; Quasilinear Utility
x2
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.
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.
Therefore an income inferior good’s Engel curve is negatively sloped.

Income Effects
In the US over last hundred years income increased many times whereas the number of kids per household went down.
Are children an inferior good?

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
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
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
p1 price offer
curve

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

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:
so
Therefore commodity 2 is a gross
complement for commodity 1.

Cross-Price Effects
p1
x1*
p1’
p1’’
p1’’’
Increase the price of
good 2 from p2’ to p2’’
and

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.

Cross-Price Effects
A Cobb- Douglas example:
so